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Hartmut Muller

Global Scaling

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the fundamentals of

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interscalar cosmology

New Heritage Publishers

Hartmut Muller

GLOBAL SCALING

the fundamentals of interscalar cosmology

Brooklyn, New York, USA
2018

This book is published and distributed in agreement with the Budapest Open Ini¬
tiative. This means that the electronic copies of the book should always be accessed
for reading, download, copying, and re-distribution for any user free of charge.

The book can be downloaded on-line, free of charge from various electronic web
libraries in the internet. To order printed copies of this book, contact the Author,
Hartmut Muller: hm@interscalar.com

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This book was typeset using the IAT^X typesetting system.

ISBN 978-0-9981894-0-6

New Heritage Publishers
Brooklyn, New York, USA

Content

Editorial Foreword.4

Preface.5

Many Questions — No Answers.6

The Power of Euler’s Number.8

The Fundamental Fractal.15

The Fundamental Metrology.19

The Fundamental Field.28

Global Scaling.33

Interscalar Cosmology.39

Nothing is Artificial in the Universe.64

Bibliography.83

Acknowledgements.86

About the Author.88

Editorial Foreword

Each fundamental scientific discovery that changed Humanity’s views
of the world did not appear out of nowhere but was built on the “shoul¬
ders” of the giants of science from before. This is also true for Global
Scaling authored by Hartmut Muller.

The natural philosophers of ancient Greece already pointed out that
all of physical reality is created according to a harmonic hierarchy which
seems to be valid for the full range of physical scales, from the very small
particles of substance to the universe as a whole.

In the 1950s, Kyril Dombrowski (1913-1997), a mathematician and
optical engineer, discovered how the minima and maxima in the rational
number distributions determine various resonance phenomena such as
the distribution of the orbits of planets and the electrons in the atom.

Benoit Mandelbrot (1924-2010), one of the great mathematicians of
the 20th century, discovered in the 1970s the key role fractal sets play
in the organisation of physical reality. Julia and Mandelbrot Sets show
endless repetitions of their fragments throughout all the complete range
of physical scales. Mandelbrot called it the fractal geometry of nature.

Hartmut Muller continued this line of research in the 1980s. He de¬
veloped the concept of Global Scaling, a theory based on the properties
of rational and transcendental numbers and continued fractions. It ex¬
plains how the distributions of numbers are not only responsible for the
structure of matter but also the dynamics of the physical, biological and
social phenomena.

Applications of Muller’s Global Scaling are so manifold and varied
that they cover all known fields of science from mathematics, physics,
and astronomy to biology, political science and economics. In this book,
many of the possible fundamental applications of Global Scaling can
only be briefly outlined and will require further detailed study, while its
main focus is placed on examples from physics and astronomy — the
structure of matter — as the professional fields of the author.

This small book is designed to be studied for years to come. We are
convinced that after reading it many young people will come to explore
the fundamentals of science. This book is truly a stimulus of thought
for future generations of research.

We are greatly honoured to be editors of this book. And it is our
duty as scientists to support this research and its author.

Pushchino, July 12, 2018 Dmitri Rabounski and Simon Shnoll

Preface

The time will come when
all people will see as I do.

Giordano Bruno

I welcome you to the book for thinking, curious, courageous and honest
people! So it might not be interesting to everyone.

But anyone looking at the world with an inquisitive mind will not
regret following me to experience the spirit of exploration of the universe
in a way that few have done before! If, for most of your life, you have
searched for certain answers, you may actually find them here...

I can promise that reading this book will not be a waste of your
time. It is the experience of a discovery that I want to share with you.

Sit back and enjoy the ride...

Valle del Sole, July 1, 2018

Hartmut Muller

Many Questions — No Answer

Did you ever ask yourself why the universe is so big? Forty thousand
billion kilometers 1 to the neighboring Alpha Centauri system, two mil¬
lion light years to the Andromeda galaxy! And, did you ever ask why
the universe is so small? A thousandth of a millimeter for a living cell,
a ten-millionth of a millimeter for a whole atom!

If you do not know the answers, you don’t need to be ashamed. Even
modern science has no plausible explanation. And, this is not an excep¬
tion, but rather a typical situation. Always when science can’t answer
simple questions, some new paradigm rises at the scientific horizon.

There are so many questions without answer, you do not believe
that? Here are some of them:

• Why is the normal resting heart rate for adults close to one beat
per second and the breathing rate close to 15 breaths per minute?

• Why does the electrical Theta activity of the brain range between
3 and 7 Hz, the Alpha activity between 8 and 13 Hz and the Beta
activity between 14 and 34 Hz?

• Why is the adult human brain mass close to 1.4 kg?

• Why does the hypophysis gland weight 500 mg?

• Why is the wavelength 280 nrn dividing ultraviolet B and C light?

• Why is the average temperature of the cosmic microwave back¬
ground radiation 2.725 K?

• Why have the Sun and the Moon, the gas giant Jupiter and the
planetoid Ceres, but also Earth and Mars similar rotation periods?

• Why have different planets as Venus and Uranus, as well as Mars
and Mercury similar surface gravity accelerations?

• Why have several planets in the Trappist 1 system the same orbital
periods as the moons of Jupiter, Saturn and Uranus?

There are many more questions like these — there are thousands.
That’s no joke. And all these questions are not even topics of theoretical
research, because in the current paradigm of science, they are considered
to be accidental.

Perhaps you will ask now — how can it be that so many questions
remain unanswered while science is dealing with black holes and dark
matter?

Forty thousand billion kilometers are 4.3 light years.

Many Questions — No Answer

A very good question. First, the emergence of highly speculative,
non-measurable entities and their exploration is a typical feature of a
conceptual crisis in natural science and a strong indicator of an upcom¬
ing profound paradigm shift. Secondly, the emergence of those entities
is a sign of psychological repression, whereby real facts are excluded
from the conscious perception and substituted by exotic and surrealis¬
tic ideas which convince you that it is vital to know how you can exit a
black hole after it has eaten you.

By the way, did you notice something that they have in common
while reading the example questions above?

Well, all the questions are about concrete measurements. And that
is precisely what today’s paradigm lacks. Known laws of nature describe
how one quantity changes in dependency on another. For example, Ke¬
pler’s third law describes how the orbital period of a planet changes
with its orbital distance. However, Kepler’s law cannot explain why
the solar system has established Jupiter’s orbital period at 11.86 years
and not 10.27 or 14.69 years. Even Newton’s gravitational law or Ein¬
stein’s theory of relativity cannot explain this. And this isn’t just a
shortcoming of astrophysics only.

In biophysics, Kleiber’s law describes how metabolic rates in mam¬
mals depend on the body mass. The law affirms that larger-bodied
species like elephants have lower mass-specific metabolic rates and lower
heart rates, compared to smaller-bodied species like mice. However, cur¬
rently there is no law known that could explain why billions of adults of
the species Homo sapiens prefer to have a heart rate of 60 - 70 beats per
minute and a breathing rate of 12 17 breaths per minute. Furthermore,
currently there is no law known that could explain why brain oscilla¬
tions of the Alpha type range between 8 and 13 Hz, of the Beta type
between 14 and 34 Hz, why all mammals have these brain frequency
ranges in common and why they coincide with Schumann-resonances.

Reading this book, you will find reasonable and precise answers to
all these and many other questions. You will also see that all these
questions have a common origin and therefore a common explanation.

Now you are going to make an amazing discovery and I’m happy to
accompany you! It feels like I was going to make this experience again
and I envy you for the moment of a pure rush of adrenaline that awaits
you and may change your life forever.

The Power of Euler’s Number

If you want to find the secrets of the universe,
think in terms of frequency and vibration.

Nikola Tesla

Ok let’s get started. Talking about measurement 1 , it results always in a
number that is the ratio of two physical quantities where one of them is
the reference quantity called unit of measurement. For example, 0.615
years, the orbital period of Venus. In this case, the orbital period of the
Earth (one year) is the unit of measurement. Therefore, the number
0.615 is the Venus-to-Earth orbital period ratio.

If this ratio were equal 1/2 or 2/3, then Venus’ orbital movement
would be in resonance with that of the Earth. In that case, periodic
interaction could progressively rock the orbital movement of both plan¬
ets and ultimately cause a resonance disaster that could destabilize the
whole solar system. Therefore, the solar system can establish only those
orbits which avoid whole number ratios.

In mathematics, ratios of whole numbers are called rational numbers.
For example, 2/3 (two-thirds) is a rational number. Besides rational
numbers, there are also irrational numbers. They cannot be represented
as a ratio of whole numbers and consequently, they should not cause
destabilizing resonance interaction. 2

For example, the square root of two \/2 = 1.414... or the golden
number 4 > = (v / 5+l)/2 = 1.618... are irrational numbers. Several
authors 3 have suggested that the Venus-to-Earth orbital period ratio
0.615 corresponds with the reciprocal golden number 1/0 = 1/1.618...=
0.618...

With reference to the solar system we may therefore expect that if
the ratio of two orbital periods is not rational but, for example equals

1 International Vocabulary of Metrology — Basic and General Concepts and As¬
sociated Terms. International Bureau of Weights and Measures, 2008.

2 Dombrowski K. Rational Numbers Distribution and Resonance. Progress in
Physics , issue 1, 65-67, 2005.

3 Pletser V. Orbital Period Ratios and Fibonacci Numbers in Solar Planetary and
Satellite Systems and in Exoplanetary Systems. arXiv: 1803.02828 (2018); Butusov
K. P. The Golden Ratio in the solar system. Problems of Cosmological Research ,
vol. 7, Moscow-Leningrad, 1978.

The Power of Euler’s Number

tj>, the orbital movements should not have resonance interaction and
should not destabilize the system.

Nevertheless, even this irrational ratio can cause a resonance disas¬
ter, because according to the third law of Kepler, the cube of the orbital
distance is proportional to the square of the orbital period. Indeed, the
square of \[2 returns the whole number 2. Even the square of the golden
number </> returns the whole number 5 after its multiplication by 2 and
removal of 1.

This is why roots of whole numbers, even being irrational, cannot
guarantee that resonance interaction will be avoided concerning all phys¬
ical quantities of the system.

Fortunately, there is another type of irrational numbers called tran¬
scendental which are not roots of whole or rational numbers. They
cannot be transformed into rational or whole numbers by addition or
multiplication and consequently, they should never provide resonance
interaction.

Indeed, planets are changing their position in space continuously.
This temporal change of the position in space is described by the ve¬
locity, a quantity called derivative. The derivative of a quantity is its
instantaneous rate of change.

Actually, the orbital velocity isn’t constant either, but increases and
decreases with the change of the orbital distance. This temporal change
of the velocity is described by an acceleration, a derivative of the veloc¬
ity. Naturally, the acceleration isn’t constant either.

If you ask me now if there is any real transcendental function that in¬
hibits resonance interaction also regarding velocities, accelerations and
other derivatives, I can give you a very positive answer. Yes, there is
one, but only one solution: it is the natural exponential function e x ,
because it is the only function that is the derivative of itself:

For x — 1 the natural exponential function e x gives Euler’s number e =
2.71828... 1

Consequently, so long as the ratio of physical quantities is given
by the natural exponential function e x , the ratios of their derivatives
will be also given by the natural exponential function e x , so that the
system remains stable even when quantities are changing. And this is
valid for any system, regardless of its complexity, because of the unique
arithmetic properties of Euler’s transcendental number e = 2.71828...

1 Maor E. e: The Story of a Number. Princeton University Press, 1994.

The Power of Euler’s Number

As we can see, the classification of real numbers, in particular the
difference between rational, irrational and transcendental numbers is
not only a mathematical task. It is also an essential aspect of stability
in complex systems. 1

Now let’s come back to our initial example of the orbital period of
Venus. How can we find out if the Venus-to-Earth orbital period ratio
is a rational, irrational or transcendental number?

Judging from the first impression, the obtained value 0.615 seems
to be a rational number, because it has a finite number of digits and
can be presented as a ratio of whole numbers: 0.615 = 123/200. On
the other hand, the circumstance that the number of digits is finite,
could be also a consequence of limited precision of measurement. In
fact, higher resolution data 2 deliver more digits, for example 0.615198
years = 224.701 days = 224 days, 16 hours and 49 minutes. Indeed,
also this value is only an average.

In reality, the sidereal orbital period of Venus is not constant, but
varies between 224.695 days = 0.615181 years and 224.709 days =
0.615220 years. According to classic models, that’s due to perturbations
from other planets, mainly Jupiter and Earth. Usually the uncertainty
is put in brackets so we can approximately write 0.61520(2) years for
the sidereal orbital period of Venus.

Let’s take another example. In 1990, the worldwide best measure¬
ments 3 of the proton-to-electron mass ratio delivered the value
1836.152701(37). In 2017, the value 1836.15267389(17) was obtained.
As you can see, not only the resolution is improved by two digits, but
also the values of some lower digits are changed. Nevertheless, the 2017
measurements do not contradict the 1990 ones, because the limits of the
2017 measurements are within the 1990 limits, confirming the hypoth¬
esis about constancy of the proton-to-electron mass ratio.

Now you can understand that it is not so simple to clarify the type
of number a measured ratio corresponds to. In general, there is no
possibility to know it for sure. However, considering the finite resolution
of any measurement, we can state that any obtained value is always
an approximation and it is very important to know the amount of its
uncertainty.

It is remarkable that approximation interconnects all types of real
numbers — rational, irrational algebraic and transcendental. In 1950,

iPanchelyuga V. A., Panchelyuga M. S. Resonance and Fractals on the Real
Numbers Set. Progress in Physics, issue 4, 48-53, 2012.

2 Venus Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov

3 Particle Data Group, www.pdg.lbl.gov

The Power of Euler’s Number

the mathematician Khinchin 1 made a very important discovery: He
could demonstrate that continued fractions deliver biunique (one-to-
one) representations of all real numbers, rational and irrational. Where¬
as infinite continued fractions represent irrational numbers, finite con¬
tinued fractions represent always rational numbers. In this way, any
irrational number can be approximated by finite continued fractions,
which are the convergents and deliver always the nearest and quickest
rational approximation.

It is notable that the best rational approximation of an irrational
number by a finite continued fraction is not a task of computation, but
only an act of termination of the fractal recursion. For example, the
golden number <j> = (\/5+l)/2 = 1.618... has a biunique representation
as simple continued fraction:

<j>= 1 +-

To save space, in the following we use square brackets to write down
continued fractions, for example the golden number <f> = [1; 1, 1, ...].
As you can see, it contains only the number 1. So long as the sequence of
denominators is considered as infinite, this continued fraction represents
the irrational number <f>. If only you truncate the continued fraction, the
sequence of denominators will be finite and you get a convergent that is
always the nearest rational approximation of the irrational number <p.

Let’s see how it works. Increasing always the length of the continued
fraction, we obtain the following sequence of rational approximations of
</>, from the worst to always better and better ones:

[ 1 ] = 1

[1; 1] = 2

[1; 1, 1] = 3/2 = 1.5

[1; 1, 1, 1] = 5/3 = 1.66

[1; 1, 1, 1, 1] = 8/5 = 1.6

[1; 1, 1, 1, 1, 1] = 13/8 = 1.625

[1; 1, 1, 1, 1, 1, 1] = 21/13 = 1.615384

[1; 1, 1, 1, 1, 1, 1, 1] = 34/21 = 1.619047

[1; 1, 1, 1, 1, 1, 1, 1, 1] = 55/34 = 1.6176470588235294117

[1; 1, 1, 1, 1, 1, 1, 1, 1, 1] = 89/55 = 1.618

1 Khintchine A. Continued fractions. University of Chicago Press, Chicago, 1964.

The Power of Euler’s Number

2,0000

0123456789

Figure 1: The approximation steps 0-9 of the golden number <j> = 1.618...
(dotted line) by continued fraction.

Figure 1 demonstrates the process of step by step approximation. As
you can see, the rational approximations oscillate around the eigenvalue
(j> of the continued fraction that is shown as dotted line. With every step
the approximation comes closer and closer to 4>, never reaching it and
describing a damped asymptotic oscillation around <f>.

By the way, in 1950 Gantmacher and Krein 1 have demonstrated
that continued fractions are solutions of the Euler-Lagrange equation for
low amplitude harmonic oscillations in simple chain systems. Terskich 2
generalized this method for the analysis of oscillations in branched chain
systems. The continued fraction method can also be extended to the
analysis of chain systems of harmonic quantum oscillators. 3

The rational approximations of the golden number </> are always ra¬
tios of neighboring Fibonacci numbers — the elements of the recursive
sequence 1, 1, 2, 3, 5, 8, 13, ... where the sum of two neighbors always
yields the following number 4 .

1 Gantmacher F. R., Krein M. G. Oscillation matrixes, oscillation cores and low
oscillations of mechanical systems. Leningrad, 1950.

2 Terskich V. P. The continued fraction method. Leningrad, 1955.

’Muller H. Fractal Scaling Models of Natural Oscillations in Chain Systems and
the Mass Distribution of Particles. Progress in Physics, 2010, issue 3, 61—66.

4 Devlin K. The Man of Numbers. Bloomsbury Publ., 2012.

The Power of Euler’s Number

As you can see, only the 10 th approximation gives the correct third
decimal of <j>. The approximation process is very slow because of the
small denominators. In fact, the denominators in the continued fraction
of <f> are the smallest possible and consequently, the approximation speed
is the lowest possible. The golden number (j> is therefore treated as the
“most irrational” number in the sense that a good approximation of <fi
by rational numbers cannot be given with small quotients.

On the contrary, transcendental numbers can be approximated ex¬
ceptionally well by rational numbers, because their continued fractions
contain large denominators and can be truncated with minimum loss
of precision. For instance, the simple continued fraction of the circle
number 7 r = 3.1415927... = [3; 7, 15, 1, 292, ... ] delivers the following
sequence of rational approximations:

[3] =3

[3; 7] = 3.142857

[3; 7, 15] = 3.14150943396226

[3; 7, 15, 1] = 3.1415929...

We can see that the 2 nd approximation delivers the first 2 decimals
correctly, and the 4 th approximation shows already 6 correct decimals.

Much like the continued fraction of the golden number (f> contains
only the number 1, a prominent continued fraction 1 of Euler’s number
contains all natural numbers as denominators and numerators, forming
an infinite fractal sequence of harmonic intervals:

e — 2 +

1 +

2 +

3 +

As Euler’s number e = 2.71828... is transcendental, it can also be
represented as continued fraction with quickly increasing denominators:

e = 1 +

1 1

-L \

6+ -

14 +

Wiu P. The Elementary Mathematical Works of Leonhard Euler. Florida At¬
lantic University, 1999, pp. 77-78.

The Power of Euler’s Number

In this way, already the 4 th approximation delivers the first 3 decimals
correctly and returns in fact the rounded Euler’s number e = 2.71828...
of 5 decimals’ resolution:

2.714285

2.7183...

This special arithmetic property of the continued fractions 1 of transcen¬
dental numbers has the consequence that transcendental numbers are
distributed near by rational numbers of small quotients.

This can create the impression that complex systems like the solar
system provide ratios of physical quantities which correspond with ra¬
tional numbers. Actually, they correspond with transcendental numbers
which are located close to rational numbers.

Only transcendental numbers define the preferred ratios of quantities
which avoid destabilizing internal resonance interaction. In this way,
they sustain the lasting stability of complex systems. At the same time,
a good rational approximation can be induced quickly, if local resonance
interaction is required temporarily.

As we have seen, among all transcendental numbers, Euler’s number
is very special, because its real power function coincides with its own
derivatives. Euler’s number allows for inhibiting resonance interaction
regarding all internal processes and their derivatives.

In the next chapter you will learn that this arithmetic property of
Euler’s number has the consequence that complex systems tend to es¬
tablish relations of quantities that coincide with values of the natural
exponential function e x for integer and rational exponents x.

Perron O. Die Lehre von den Kettenbriichen. 1950.

The Fundamental Fractal

There is one fundamental
cause of all effects.

Giordano Bruno

Thanks to Khinchin’s discovery, any real number can be represented as
a continued fraction. Now let’s apply it to the real argument x of the
natural exponential function e x itself:

x = [n 0 ;ni,n 2 ,... ,n k ]

All denominators ni, n 2 ,..., nu of the continued fraction including the
free link no are integer (positive and negative whole) numbers. All
numerators equal 1. The length of the continued fraction is given by
the number k of layers.

The canonical form (all numerators equal 1) does not limitate our
conclusions, because every continued fraction with partial numerators
different from 1 can be transformed into a canonical continued frac¬
tion using the Euler equivalent transformation 1 . With the help of the
Lagrange 2 transformation, every continued fraction with integer denom¬
inators can be represented as a continued fraction with natural denom¬
inators that is always convergent 3 .

Now let’s look at the fractal distribution of rational eigenvalues of
finite continued fractions. The first layer is given by the truncated after
ni continued fraction:

x = [n 0 ;ni] = n 0 -1-

n i

For the beginning we take no = 0. The denominators ni follow the
sequence of integer numbers =bl, ±2, ±3 etc. The second layer is given

1 Skorobogatko V. Ya. The Theory of Branched Continued Fractions and math¬
ematical Applications. Moscow, Nauka, 1983.

2 Lagrange J. L. Additions aux elements d’algebre d’Euler. 1798.

3 Markov A. A. Selected work on the continued fraction theory and theory of
functions which are minimum divergent from zero. Moscow-Leningrad, 1948.

The Fundamental Fractal

by the truncated after n 2 continued fraction:

x = [n 0 ;ni,n 2 ] = n 0 H-—

n 1 H-

n 2

Figure 2 shows the first and the second layer in comparison. As you
can see, reciprocal whole numbers ±1/2, ±1/3, ±1/4,... are the attrac¬
tor points of the distribution. In these points, the distribution density
always reaches a local maximum. As well, you can recognize that whole
numbers 0, ±1,... are the main attractors of the distribution.

Now let’s remember that we are observing the fractal distribution of
rational values x = [no; ni, n 2 ,..., Uk\ of the real argument x of the nat¬
ural exponential function e x . What we see is the fractal distribution of
transcendental numbers of the type e x on the natural logarithmic scale!
And, we can see that near whole number exponents the distribution
density of these transcendental numbers is maximum!

Consequently, for integer exponents, the natural exponential func¬
tion e x defines attractor points of transcendental numbers and create
islands of stability! Let’s write them down:

e° = 1; e 1 = 2.718...; e 2 = 7.389...; e 3 = 20.085...; e 4 = 54.598...;
e 5 = 148.413...; e 6 = 403.428...

Figure 2 shows that these islands are not points, but ranges of sta¬
bility. Integer number exponents like 0, ±1, ±2, ±3,... are attractors
which form the widest ranges of stability. Half exponents ±1/2 form
smaller islands, one third exponents ±1/3 form the next smaller islands
and one fourth exponents ±1/4 form even smaller islands of stability.

In this way, the natural exponential function e x of the rational ar¬
gument x = [no;ni,n 2 , • ■ •, ti*] generates the set of preferred ratios of
quantities which provide the lasting stability of real processes and struc¬
tures regardless of their complexity. This is a very powerful conclusion,
as we will see in the following.

For rational exponents, the natural exponential function is always
transcendental. 1 Increasing the length of the continued fraction, the
density of the distribution of transcendental numbers of the type e x on
the natural logarithmic scale is increasing as well. In fact, nearly every
irrational number is transcendental, and all irrational numbers together
form a continuum.

Hilbert. D. Uber die Transcendenz der Zahlen e und tx. Mathematische Annalen,
Bd. 43, 216-219, 1893.

The Fundamental Fractal

Nevertheless, their distribution is not homogeneous, but fractal. Ap¬
plying continued fractions and truncating them, we can represent the
exponents of the natural exponential function e x as rational numbers
and make visible their fractal distribution.

Here I would like to underline that the application of continued frac¬
tions doesn’t limit the universality of our conclusions, because continued
fractions deliver biunique representations of all real numbers including
transcendental.

Therefore, the fractal distribution of eigenvalues of the natural ex¬
ponential function e x of the real argument x, represented as simple con¬
tinued fraction, is an inherent characteristic of the number continuum.
This characteristic we call the Fundamental Fractal (FF). 1

Let us remember now that in physical applications, the natural ex¬
ponential function e x of the real argument x is the ratio of two physical
quantities where one of them is the reference quantity called unit of
measurement. Therefore, now we can rewrite our equation:

ln(X/Y) = [n 0 ;ni,n 2 ,.. .,n k ]

where X is the measured physical quantity and Y the unit of measure¬
ment; In is the natural logarithm.

Now let’s apply this knowledge to our first example of the Venus-
to-Earth orbital period ratio 0.61520(2). In this case, X = 0.61520(2)
years and Y = 1 year. Let’s calculate the natural logarithm of the ave¬
rage: ln(0.6152) = —0.49. We can see that this logarithm is close to
— 1/2. The deviation is only 0.01. Consequently, the Venus-to-Earth
orbital period ratio is close to an attractor point of the FF. To reach
this attractor point that is the center of a local island of stability, seems
that Venus has to increase its orbital velocity slightly.

Indeed, our calculation did not consider all uncertainties in the
Venus-to-Earth orbital period ratio. As we have found out, the un¬
certainty of this ratio appears as consequence of periodic variations in
the orbital movement of Venus. Certainly, this is valid not only for
Venus, but for all celestial bodies. Also the orbital movement of the
Earth is not constant.

1 Miiller H. Scale-Invariant Models of Natural Oscillations in Chain Systems and
their Cosmological Significance. Progress in Physics, issue 4, 187-197, 2017.

The Fundamental Metrology

The Eternal is number, measure,
limit without limit, end without end.

Giordano Bruno

In this context, the question arises whether there is some kind of “ab¬
solutely” stable process in the universe?

In fact, such processes do exist. Historically one of them was discov¬
ered as cathode rays and named “electron”, another as nucleus of the
hydrogen atom and named “proton”.

The lifespans of the proton and electron 1 surpass everything that is
measurable, exceeding 10 30 years. No scientist ever witnessed the decay
of a proton or an electron. Proton and electron form stable atoms, the
structural elements of matter.

The exceptional stability and uniqueness of the electron and proton
predispose their physical characteristics to be treated as natural and
fundamental units of measurement. Table 1 on the next page shows
the basic set of electron and proton units (c is the speed of light in a
vacuum, h is the Planck constant, ks is the Boltzmann constant) that
we call the Fundamental Metrology.

The Fundamental Metrology is completely compatible with Planck
units. Originally proposed in 1899 by Max Planck, they are also known
as natural units, because the origin of their definition comes only from
properties of nature and not from any human construct. Natural units
are based only on the properties of space-time.

Max Planck wrote that these units, “regardless of any particular
bodies or substances, retain their importance for all times and for all
cultures, including alien and non-human, and can therefore be called
natural units of measurement”. 2

Richard Feynman was a student in Princeton in the spring of 1940,
when during a telephone conversation, his professor of physics John
Wheeler shared with him an idea of cosmological significance. In his
speech at the receipt of the Nobel Prize, Feynman recounted this story

1 Steinberg R. I. et al. Experimental test of charge conservation and the stability
of the electron. Physical Review D., 1999, vol. 61 (2), 2582—2586.

2 Max Planck. Uber Irreversible Strahlungsvorgange. Sitzungsbericht der Konig-
lich Preuftischen Akademie der Wissenschaften , 1899, vol. 1, 479—480.

The Fundamental Metrology

PROPERTY

ELECTRON

PROTON

rest mass m

9.109383- 10" 31 kg

1.672622- 10" 27 kg

energy E = me 2

0.5109989 MeV

938.27208 MeV

temperature T = E/fcs

5.9298446 ■ 10 9 K

1.08881- 10 13 K

frequency u> = E/ft

7.763441 ■ 10 2 ° Hz

1.42 5 4 86-10 24 Hz

oscillation period t = 1/w

1.288089- 10" 21 s

7.01515 - 10' 25 s

wavelength A = c/uj

3.861593- 10" 13 m

2.103089-10 -16 m

acceleration a = cco

2.327421 ■ 10 29 ms -2

4.2735 • 10 32 ms -2

Table 1: The Fundamental Metrology. Physical characteristics of proton and
electron. Data taken from Particle Data Group, www.pdg.lbl.gov

as follows: “Feynman,” Wheeler said, “I know why all electrons have the
same charge and the same mass.” “Why?” Feynman asked. “Because,”
Wheeler replied, “they are all the same electron!”

In this book we treat the proton and electron as the “metronomes of
the universe”, as fundamental clocks which are synchronizing the whole
universe. Here arises a question: What is the source of their exceptional
stability?

In fact, the proton-to-electron ratio 1836.15267389(17) is a funda¬
mental constant 1 and it has the same value for frequencies, oscillation
periods, wavelengths, accelerations, energies and masses.

In standard particle physics, the electron is stable because it is the
least massive particle with non-zero electric charge. Its decay would
violate charge conservation. The proton is stable, because it is the
lightest baryon and the baryon number is conserved. Indeed, this answer
only readdresses the question. Why then is the proton the lightest
baryon? To answer this question, scientists believe in the existence of
non-observable entities — the quarks...

Now hold on tight: It may be that the source of the exceptional
stability of the proton and electron is the number continuum, more
specifically, the proton-to-electron ratio itself is caused by the FF! In
fact, the natural logarithm is close to seven and a half:

In ( Wproton = In (1836.15267389) ~ 7 + ]-

\ electron / "

1 Physical constants. Particle Data Group, www.pdg.lbl.gov

The Fundamental Metrology

As a consequence, the proton FF is complementary to the electron FF,
because integer logarithms of the proton FF correspond to half log¬
arithms of the electron FF and vice versa, so that the scaling factor
e 1 / 2 = y/e = 1.64872... connects attractor points of proton stability
with similar attractor points of electron stability in alternating sequence.
Figure 3 on the next page demonstrates this situation.

In the bottom we see the proton FF and in the top the electron
FF. Both are represented at the first layer only, so we can see clearly
that they have in common only the attractor points ±1/2, ±1/3 and
±1/4. In these attractor points, proton stability is supported by electron
stability, so we can expect that they are preferred in complex systems.

By the way, not only the proton-to-electron ratio follows the FF, but
also the ratios of other elementary particles do so, even if their lifespans
are very short. As table 2 shows, the logarithms of fundamental particle
ratios are always close to integer or half values.

PARTICLE

mass m, MeV/c 2

In (m/m e )

In (m/m e )-FF

H-boson

125090

12.408

[12;2]

-0.092

Z-boson

91187.6

12.092

[12]

0.092

W-boson

80385

11.966

[12]

-0.034

neutron

939.565379

7.517

[7;2]

0.017

proton

938.27208

7.515

[7;2]

0.015

electron

0.51099894

0.000

[0]

0.000

Table 2: Fundamental particles and the correspondence of their mass ra¬
tios with FF-attractors of stability. Data taken from Particle Data Group,
www.pdg.lbl.gov

We know already that the islands of stability in the FF are not points
but ranges. Integer exponents like 0, ±1, ±2, ±3, ... are attractor
points which form the largest islands of stability. Half exponents ±1/2
form smaller islands, one third exponents ±1/3 form the next smaller
islands and one fourth exponents ±1/4 form even smaller islands.

Therefore, we can expect that complex systems first occupy the main
islands of stability which correspond with integer or half exponents, then
the next smaller islands are occupied which correspond with one third
exponents and finally those of the one fourth exponents.

Applying this rule to the analysis of measurements we can study the

The Fundamental Metrology

fractal hierarchy of complex systems and understand their formation.

In astrophysics, it allows for the prediction of orbits of missing ce¬
lestial bodies in the solar system as well as exoplanets. In biophysics
and astrobiology, it allows for studying the interscalar embedding of
biological functions in astrophysical processes. In geophysics and plan¬
etology, it allows for the prediction of the lithospheric and atmospheric
stratification on various planets (see references on pp. 83-84).

Now let’s apply the metric characteristics of proton and electron
the Fundamental Metrology — to the analysis of measurements. Let’s
take our famous example of Venus’ orbital period. Let’s calculate the
natural logarithm of the ratio of Venus’ sidereal orbital period to the
electron oscillation period:

f T Venus \ = / 224.701 • 86164 s \
\27t • T electro J \27t • 1.288089 • 10” 21 s)

63.04

We can see that this logarithm is close to the integer 63. The devia¬
tion is only 0.04. Consequently, Venus’ orbital period is near the main
attractor E[63] of electron stability 1 . This result confirms our early con¬
clusion that Venus has to increase its orbital velocity slightly to reach
the attractor point of stability in the FF.

Also the orbital period of the Earth corresponds with a main attrac¬
tor of stability, but relative to the proton oscillation period:

( T Earth \ , ( 365.2564 • 86164 s \

V 27r • Voton ) n V 27r • 7.015150 • 10- 25 s )

71.05

Probably, also the Earth will increase its orbital velocity slightly. Jupi¬
ter’s sidereal orbital period 2 coincides perfectly with the main attractor
E[66] of electron stability:

/ T ju piter \ = ln / 4332.59 • 86164 s \

\27T • Telectron / \ 2 tT • 1.288089 • 10" 21 S )

66.00

Jupiter is the largest and heaviest planet in the solar system and fortu¬
nately, Jupiter’s orbit is perfectly positioned in the FF. Thank God!

Jupiter’s rotation period of 9.84 hours coincides with the same main
attractor P[66], but of proton stability:

ln ( =h, ( 9 ' 84 ' 3600 8 )

Uproton J \7.015150 • 10~ 25 S )

66.09

^^Here and in the following we use the letter E for electron stability and the letter
P for proton stability.

2 Jupiter Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov

The Fundamental Metrology

When the sidereal rotation period of Jupiter slows down to P[66] = 9
hours, the orbital-to-rotation period ratio of Jupiter can be described
by the equation:

Jupiter ,, T electron
- = 27T-

T Jupiter r proton

By the way, the rotation period 9.074 hours of the planetoid Ceres fits
perfectly with the same attractor P[66].

Although the rotation of Venus is reverse, its rotation period of
5816.66728 hours fits with the main attractor E[65]:

f Venus \ /5816.66728 • 3600 s\

V Electron ) ~ V 1.288089 ■ 10~ 21 S )

64.96

The sidereal rotation period of Mars is 24.62278 hours and coincides
perfectly with the main attractor P [67]:

/ 7* Mars \
\ ^proton /

/ 24.62278 • 3600 s \
V 7.015150 • 10- 25 s )

67.01

Naturally, Earth’s rotation period 23.93444 hours coincides with the
same attractor P[67]. The sidereal rotation period of Mercury is 1407.5
hours and coincides with the main attractor P [71]:

Mercury \ h] / 1407.5 • 3600 s \
11 V 7-proton )~ “ V 7.01515 ■ 10” 25 S )

71.05

The sidereal rotation period of Neptune is 16.11 hours and coincides
with the main attractor E[59]:

( T Neptune
^electron

16.11-3600 s \
1.288089 • 10~ 21 s)

59.07

The rotation periods 1 of Saturn (10.55 h), Uranus (17.24 h) and Pluto
(152.875 li) coincide with the sub-attractors E[59;—3], P[67;—3], E[61;3]
respectively.

Figure 4 shows how the orbital periods of planets and planetoids of
the solar system are distributed in the FF. We can see that the majority
of planets has occupied main attractors of electron stability.

The Earth is the only planet that occupies a main attractor of proton
stability. Mercury, Mars, Eris and Neptune occupy sub-attractors of the
same type [no; ±3].

X NASA Space Science Coordinated Archive, www.nssdc.gsfc.nasa.gov

The Fundamental Metrology

Not only the solar system, but also exoplanetary systems like Trap-
pist 1 or Kepler 2 follow the FF. Also exoplanetary orbits are positioned
close to attractor points of proton or electron stability.

It is remarkable that the orbits of Trappist lb, c, d and e corre¬
spond with main attractors. This is also valid for Kepler 20b, d and
e and for many other exoplanetary systems we do not discuss in this
book. Therefore, their orbital periods can coincide with those in the
solar system. For example, planets in the Trappist 1 system have the
same orbital periods as have the moons of Jupiter, Saturn, Uranus and
Neptune, as shows figure 5.

The origin of the FF is the number continuum. Consequently, not
only planetary systems follow the FF, but everything in the universe
does so. Naturally, biological processes are not an exception. For in¬
stance, the average adult human relaxed heart rate 3 of 60-70 beats per
minute is close to the main attractor E[—48] of electron stability:

( 66/60 Hz \

v 7.763441 • 10 20 Hz )

The average adult human relaxed breathing 4 rate of 13-17 breaths per
minute is close to the main attractor P[—57] of proton stability:

l f U breathing \ _ , / 15/60 Hz \ _

V ^ proton ) \1.425486 • 10 24 Hz/

The EEG frequency ranges 5 of Theta (3-7 Hz), Alpha (8-13 Hz) and
Beta (14-33 Hz) brain activity follow precisely the FF:

The lower Theta limit of 3 Hz coincides with the main attractor
E[—47], the Theta-Alpha boundary of 7-8 Hz coincides with E[—46],
the Alpha-Beta boundary of 13-14 Hz coincides with P[—53] and the
upper Beta limit of 33 Hz fits perfectly with the main attractor P[—52]
of proton stability.

^ heart
^ electron

= In

billon M. et al. Seven temperate terrestrial planets around the nearby ultracool
dwarf star TRAPPIST-1. Nature , vol. 542(7642), 456-460, 2017.

2 Hand E. KEPLER discovers first Earth-sized exoplanets. Nature.com , 20 Dec.
2011 .

3 Spodick D. H. Survey of selected cardiologists for an operational definition of
normal sinus heart rate. The American J. of Cardiology , 1993, vol. 72 (5), 487-488.

4 Ganong’s Review of Medical Physiology (23 rd ed.), p. 600.

5 Tesche C. D., Karhu J. Theta oscillations index human hippocampal activation
during a working memory task. PNAS , vol. 97, no. 2, 2000.

The Fundamental Metrology

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The Fundamental Field

There is in the universe
neither center
nor circumference.

Giordano Bruno

Until now we did apply the FF to the analysis of frequencies and oscil¬
lation periods. Now let’s calibrate the FF on the proton and electron
wavelengths and apply the spatial projection of the FF to the analysis
of sizes and distances.

The number of layers of the FF is not limited. Therefore, in each
point of the space-time a scalar (real number) is defined — the eigen¬
value of the FF. In this way, the FF creates a fractal scalar field, the
Fundamental Field. 1

Figure 6 shows the linear spatial 2D-projection of the first layer
of the Fundamental Field e x for x = no + 1/ni in the interval — 1 <
x < 1. Figure 2 on page 17 shows the same interval in the logarithmic
representation. Figure 7 shows the linear 2D-projection of the FF with
both proton and electron attractors of stability.

The Fundamental Field is the spatio-temporal projection of the Fun¬
damental Fractal. For both we use the abbreviation FF. The connection
between the spatial and temporal projections of the FF is given by the
speed of light in a vacuum c = 299792458 m/s. The constancy of c
makes both projections isomorphic, so that there is no arithmetic or
geometric difference. Only the units of measurement are different.

Figures 6 and 7 show the spatial 2D-projection, but in reality the
FF is 3-dimensional, a sphere with fractal layers inside like an onion.

At each layer, the potential energy of the Fundamental Field is con¬
stant, therefore the layers are called equipotential surfaces. The poten¬
tial difference defines a gradient, a vector directed to the center of the
field that causes a central force of attraction. Indeed, the Fundamental
Field is fractal so that the gradient isn’t always directed to the center,
but exposes the internal fractality of the FF.

1 Muller H. Quantum Gravity Aspects of Global Scaling and the Seismic Profile
of the Earth. Progress in Physics, issue 1, 41—45, 2018.

The Fundamental Field

Figure 6: The equipotential surfaces of the Fundamental Field in the linear
2D-projection for k = 1.

Considering its arithmetic origin, we postulate that all types of phys¬
ical interaction including the electromagnetic, nuclear and gravitational,
originate from the FF. In view of this, they differ only in scale.

What about the field source, the electrical charge, for instance? The
appearance of a field source is only an effect of linear observation. You
can recognize this effect in figure 6. Whereas the logarithmic pattern of
the FF is the same in all scales, its linear density increases exponentially
in the direction of smaller scales, creating the effect of an existing field
source. In contrast, in the direction of larger scales, the linear density
is decreasing, creating the effect of an accelerating expansion of the FF.

Reading this, aren’t you reminded of something? Right, we are
reminded of the Big Bang model of an expanding universe! We realize
that the expansion of the universe is only an effect of linear observation
that is difficult to explain if you don’t know the FF.

Consequently, in the solar system, in the Galaxy or in any other
complex system where internal resonance interaction has to be avoided
for reasons of stability, the involved physical fields should expose the
inherent structure of the FF. In fact, analyzing distances in the solar

The Fundamental Field

Figure 7: The Fundamental Field with equipotential surfaces of both proton
and electron attractors of stability in the linear 2D-projection for k = I.

system, we can see that the orbits of planets and moons coincide with
equipotential surfaces of the FF, because movement along an equipo¬
tential surface requires no work.

For example, the orbital distance 1 of Venus coincides with the main
equipotential surface E[54] of electron stability:

1 f R Venus \ / 1.08939 • 10 11 m \

n \ A electron / H \3-861593 • 10“ 13 m)

where Ry enus = 0.723332 AU = 1.08939 • 10 11 m is the semi-major
axis of Venus’ orbit, A electron = 3.861593 • 10~ 13 m is the Compton
wavelength of the electron (see table 1 on page 20). Earth’s orbital
distance coincides with the equipotential surface E[54;3]:

In = In f 1-49598023-10^ m N = M + 1

\ ^ electron J V 3.861593 • 10" 13 m J 3

The mean orbital distance of Jupiter coincides with the main equipo-
1 Venus Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov

The Fundamental Field

tential surface E[56] of electron stability:

, / R Jupiter \ . ( 7.7857 • 10 11 m \

n U electron/ U 8 61593 • 10" 13 mj

Now you can understand that the knowledge of the FF opens the pos¬
sibility to develop a completely new vision of the solar system. The
origin of this vision is the number continuum and therefore, it is not
only precise, but universal and applicable to all systems in the Galaxy.

The spatial projection of the FF determines not only the orbital
systems, but also the sizes of stars, planets and moons. For example, the
radius of Sirius A photosphere 1 coincides with the main equipotential
surface P[57] of proton stability:

, ( r Sirius \ . ( 1.19155 • 10 9 m \

VAprotoJ n V 2-103089 -10-16 m)

The radius of the Sun’s photosphere coincides with the main equipoten¬
tial surface E[49] of electron stability:

= ( 6-86407-IQ*,, \ = 49

\ ^ electron ) v 3 - 861593 ' 10-13 m /

The radius of the photosphere is considered as the Sun’s “surface” in
the definition of the surface gravity acceleration of the the Sun 2 that is
actually 274 m/s 2 , it is 28 times stronger than gravity on the Earth’s
surface. Sun’s surface gravity coincides with the main attractor E[—62]:

Inf 9S "° j=lnf - 274 m/s ' ,

V a electron J \ 2.327421 • 10 29 nr/s

All units of measurement we are using are taken from the Fundamental
Metrology, see table 1 on page 20. Jupiter’s surface gravity coincides
with the main attractor P[—72] of proton stability:

in (= In ( 24 - 8 m/s2 ^

\a proton J y 4.273500 • 10 32 m/s 2

1 Liebert J. et al. The Age and Progenitor Mass of Sirius B. The Astrophysical
Journal , vol. 630(1), 69-72. arXiv:astro-ph/0507523.

2 Sun Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov

The Fundamental Field

Venus and Uranus have the same surface gravity that coincides with the
main attractor P[—73] of proton stability:

ln /5Venus\
\ & proton /

( 8.8 m/s 1 2 * \

\ 4.273500- 10 32 m/s 2 J

= -73

Mars and Mercury have the same surface gravity that coincides with
the sub-attractor E[—66;—3] of electron stability:

9 Mars
& electron

( 3.7 m/s 2 \

y 2.327421 • 10 29 m/s 2 J

- 66 -

Earth’s surface gravity coincides with the sub-attractor E[—65;—3]:

Inf 9E " h Min ( - 9 ' 8 m/s2 -A =-65-i

V a electron J \ 2.327421 • 10 29 m/s' J 3

It is always the same logic: If it is vital for the system that some ratio
of quantities remains stable, for example the orbital period of Jupiter
in relation to the oscillation periods of proton and electron, then this
ratio should be as close as possible to an FF-attractor point of stability.

Having this position in the FF, the ratio of quantities doesn’t have
to fit precisely with an attractor point, because near an attractor point
this ratio is “swimming” in a pool of transcendental numbers.

Here I want to underline that the Fundamental Field does not prop¬
agate, it is omnipresent. The Fundamental Field is the spatio-temporal
projection of the Fundamental Fractal that is an inherent feature of the
number continuum. The FF causes the fractality of space-time.

In physics, only field distortions (waves or currents), not the fields
themselves have propagation speeds. In astronomic calculations, gravi¬
tation is traditionally considered as being instantaneous. First Laplace 1
demonstrated that gravitation as field does not propagate with the speed
of light c. Modern estimations 2 confirm a lower limit of 2 • 10 10 c.

Also, the quantization of orbital systems is not a random solution.
The solution is given a priori and it is omnipresent. Therefore, we find
that orbital quantization follows the FF also in exoplanetary systems
(see figure 5 on page 27).

1 Laplace P. Mechanique Celeste. 1825, pp. 642-645.

2 Van Flandern T. The Speed of Gravity — What the Experiments Say. Physics

Letters A, vol. 250, 1-11, 1998.

Global Scaling

Anything we take in the universe
has in itself that which is All in All.

Giordano Bruno

The Fundamental Fractal is of pure mathematical origin, and there is no
particular physical mechanism required as creator of the Fundamental
Field. It is all about numbers as ratios of physical quantities which
can provoke destabilizing resonance or inhibit it. In this way, the FF
concerns all repetitive processes which share at least one characteristic
— the frequency.

A general resonance condition is given by rational frequency ratios.
This condition does not compellingly cause resonance, but increases
dramatically the probability of its occurrence. Primarily, the FF defines
the distribution of those frequency ratios that inhibit resonance.

In the case of quantum oscillators, the FF defines also the ratios of
wavelengths, velocities, energies, masses and other physical quantities
which inhibit resonance and in this way, support lasting stability. It is
because many physical characteristics of quantum oscillators are con¬
nected with their frequency by fundamental constants — the speed of
light in a vacuum and the Planck constant.

Within the ocean of quantum oscillators in the universe, there are
only two of exceptional stability — electron and proton. They form
atoms — the stable elements of the universe. You remember that the
proton-to-electron ratio satisfies the condition of main attractors of sta¬
bility in the FF. This is why the FF affects everything in the universe
and is of cosmological significance. Probably, the FF is some kind of
“matrix of the universe”.

Indeed, the electron and the proton are not the ultimate sources,
but stability nodes of the FF. The spatial and temporal distribution of
these stability nodes is determined by Euler’s number. Already Paul
Dirac 1 mentioned that “... whether a thing is constant or not does not
have any absolute meaning unless that quantity is dimensionless”.

By the way, in German, “knowing from the FF” means that you
know something by heart and you can do it “on the fly”, because you

Dirac P. A. M. The cosmological constants. Nature, vol. 139, 1937.

Global Scaling

got not only a single aspect, but also all the following pages of the topic.
The term comes from Latin “ex forma, ex functione”. In Italy, “ff” has
various historical meanings: “fiat fiat” (imperative order), “fortissimo”
(powerful) and “finissimo” (very thin). I think that even these sayings
describe some features of our FF.

Now I would like to guide your attention to a very important feature
of the FF. As you already know, in the logarithmic representation, the
main attractors show an equidistant distribution (see figure 4 on page
25). Neighboring main attractors of the same calibration (E or P) are
always separated by one unit of the natural logarithm. Consequently,
if one main attractor is known, all the others can be calculated simply
by multiplication with Euler’s number. This feature of the FF is called
“scale invariance” or “scaling”.

Consequently, it is sufficient to know one metric characteristic of
the electron or proton to calculate the complete FF in all scales of the
universe with all attractors and sub-attractors of electron or proton
stability. This feature we call “Global Scaling”. 1

Already in 1795, Karl Friedrich Gauss discovered scaling in the dis¬
tribution of prime numbers. As natural phenomenon, scaling was dis¬
covered probably first in biophysics by Gustav Fechner 2 and Ernst We¬
ber. Then in seismology, Beno Gutenberg and Charles Richter 3 have
shown that there exists a logarithmic invariant (scaling) relationship
between the energy (magnitude) and the total number of earthquakes
in a given region and time period.

In the sixties, Richard Feynman and James Bjorken 4 discovered scal¬
ing in particle physics. In the eighties, the scaling exponent 3/2 was
found in the distribution of particle masses by Valery A. Kolombet. 5 In
the last 40 years many studies were published which show that scaling
is a widely distributed phenomenon. 6

Reading this book, you will learn that Global Scaling is a universal
characteristic of organized matter and criterion of stability. As we have
already seen, Global Scaling is a forming factor of the solar system.

1 Miiller H. Scale-Invariant Models of Natural Oscillations in Chain Systems and
their Cosmological Significance. Progress in Physics, issue 4, 187-197, 2017.

2 Fechner G. T. Elemente der Psychophysik, Bd. 2, 1860.

3 Gutenberg B., Richter C. F. Seismicity of the Earth and associated phenomena.
Princeton University Press, 1954.

4 Feynman R. P. Very high-energy collisions of hadrons. Phys. Rev. Lett., vol. 23,
1415, 1969; Bjorken J. D. Phys. Rev. D, vol. 179, 1547, 1969.

’Kolombet V. Macroscopic fluctuations, masses of particles and discrete space-
time. Biofizika, 1992, vol. 36, 492—499.

®Barenblatt G. I. Scaling. Cambridge University Press, 2003.

Global Scaling

In this book you will learn that Global Scaling forms also the fractal
structure of the Earth’s interior and of planetary atmospheres.

The FF defines the fractal hierarchy of attractors which are islands
of stability in very different scales - from the subatomic to the galactic.
Now let’s come back to the first question I asked at the beginning of
this book. Why is the universe so large and at the same time so small?
Naturally, you already know the answer: It is because of the logarithmic
scale invariance of the FF. Let us take an example. Hydrogen (protium)
is the smallest atom and its atomic radius 22 pm coincides with the main
attractor E[4] of electron stability:

i f r hydrogen \ _ . / 2.2 • 10 11 1U

V A electron / \ 3.861593 • ICG 13 m

Adding 12 units of the natural logarithm we find the attractor E[16] =
3.4 fim that is occupied by the smallest living cell, the mycoplasma:

in ( r 7 coplasma ') = In ( 3 - 4 ’ 1Cr6 _ m \ = 16

J V 3 - 861593 • 10-13 n v

Adding another 12 logarithmic units we find the attractor E[28] = 55
cm, the body length of a newborn.

The same scale-difference of 12 natural logarithmic units divides the
scale of the Galaxy P[84] and the scale of the solar system P[72] that
appears in the Galaxy like an atom in a living cell. On the logarithmic
scale it is always the same distance, but in linear space-time it can be
subatomic or interstellar.

The FF neither expands nor condenses into a point. The FF is of
arithmetical origin and so, it is an eternal constant that forms the uni¬
verse in all scales. Global Scaling is the conceptual basis of interscalar
cosmology.

Whereas standard cosmology 1 studies the large-scale structures and
dynamics in the universe to understand its origin, evolution and “ulti¬
mate fate”, the basis of interscalar cosmology is the study of the universe
in all scales, considering the FF as universal matrix of stability.

Interscalar cosmology considers that the apparent dominance of large
scale dynamics in the universe is only a scaling-effect of observation.

Established systems like atoms, living cells, organisms, planetary
systems or galaxies are always realizations of the same matrix - the
FF. In the universe there are no more or less important scales.

1 Ellis G. Issues in Philosophy of Cosmology. arXiv:astro-ph/0602280v2, 2006.

Global Scaling

That’s why it is difficult to understand the nature of the universe
considering only large scales. Naturally, such cosmological models can’t
be considered as complete and their affirmations about the origin, evo¬
lution and “ultimate fate” of the universe should be perceived with
healthy scepticism. Let’s see together a prominent example.

Observing the sky with a traditional optical telescope, the space be¬
tween stars and galaxies only seems completely dark. Actually, sensitive
radio telescopes receive a faint background noise, or glow, that is not
associated with any star, galaxy, or other object.

This glow is strongest in the microwave region of the radio spectrum.
Accidentally discovered in 1964 by the American radio astronomers
Arno Penzias and Robert Wilson, it was named cosmic microwave back¬
ground radiation (CMBR).

In Big Bang cosmology, the CMBR is interpreted as a remnant from
an early stage of the observable universe. According to this theory,
when the universe was young, stars and planets didn’t exist yet, and it
was denser, much hotter, and filled with a uniform glow from a hot fog
of hydrogen plasma. As the universe expanded, both the plasma and
the radiation filling it grew cooler.

Admittedly, there are alternative models 1 in development proposing
explanations for the CMBR which do not implicate standard cosmo¬
logical scenarios. However, traditionally CMBR data is considered as
critical to cosmology since any proposed model of the universe must
explain this radiation.

Within Global Scaling, there is no need for a hot prehistory of the
universe, no need for any Big Bang and no need for after-cooling. In
short, the CMBR is nothing special or out of the ordinary. It is of the
same subatomic origin as any electromagnetic radiation in the universe.

If this process is stable, it should correspond with an attractor of
the FF. In fact, the average temperature 2.725 Kelvin of the CMBR 2
corresponds to the main attractor P[—29] of proton stability:

/T C mbr\ j/ 2.725 K \
\ T proton / 11 \ 1.08881 • 10 13 K J

-29.015

and coincides perfectly with the attractor E[—21;—2] of electron stabil-

1 Lopez-Corredoira M. Non-standard models and the sociology of cosmology. Sci¬
ence Direct, Studies in History and Philosophy of Modern Physics, vol. 46, Part A,
May 2014, pp. 86—96.

2 Fixsen D. J. The Temperature of the Cosmic Microwave Background. The
Astrophysical Journal, vol. 707(2), 916—920. arXiv:0911.1955, 2009.

Global Scaling

ity:

/ T cmbr \ , ( 2.725 K \

V T electron / " \ 5.9298446 • 10 9 k;

-21.50

By the way, the average temperature 5000 K of the Sun 1 (corona) co¬
incides with the same attractor P[—21;—2], but of proton stability:

( T Sun \ f 5000 K \
\ T proton / 11 'v 1.08881 • 10 13 K )

-21.50

In this way, Global Scaling analysis shows that both processes are in¬
terconnected. This connection could indicate that the CMBR and the
solar radiation are of the same origin. Actually, the Planck satellite,
even though orbiting the Earth outside the atmosphere, is still deeply
inside the heliosphere. In fact, global asymmetry in the CMBR has
been reported 2 that is aligned with the plane of the solar system.

Knowing the FF, it is clear that there is no isotropic process in the
universe, and it isn’t surprising that this is valid also for the CMBR.
Increasingly precise data provided by the WMAP and Planck missions
made this anisotropy visible.

In contrast to conventional cosmology, interscalar cosmology is not
based on the study of the universe only in largest scales. On the con¬
trary, Global Scaling concerns the stability of any process in any scale
of the universe.

The FF defines the ratios of quantities which preserve processes and
structures from destructive internal resonance. As you already know,
within Global Scaling, resonance conditions can be expressed in terms
of frequency ratios or ratios of any other metric process characteristics,
because Global Scaling deals with stable quantum oscillators — the
proton and electron.

In quantum physics, the Boltzmann constant converts energy ra¬
tios into ratios of temperatures (see table 1 on page 20). In this case,
frequency ratios can be also expressed as ratios of temperatures, natu¬
rally only if there is no significant dependency on other thermodynamic
parameters.

Fortunately, the melting points of several substances do not show
strong dependency on pressure and other environmental conditions. For
pure chemical elements, the melting point is identical to the freezing
point and remains constant throughout the melting process. Therefore,

1 Sim Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov

2 Santos L. Influence of Planck foreground masks in the large angular scale quad¬
rant CMB asymmetry. arXiv:1510.01009vl, 2015.

Global Scaling

we can expect that the melting points of pure chemical elements cor¬
respond with attractors of proton or electron stability. You remember
that attractor points are points of change, where compression switches
to decompression and vice versa. This change can cause the transition
from solid to liquid state, for example.

In fact, the melting 1 point 0.955 K of helium 4 (under high pressure)
coincides with the main attractor P [—30]:

, ( T He \ ( 0-955 K \ _ on

1 U proton ) n [ 1.08881 -10 13 kJ

The melting point 14 K of hydrogen (protium) coincides with the main
attractor E[— 20], the melting points 19 K of deuterium and 20 K of
tritium coincide with the main attractor P[— 27] and the melting point
55 K of oxygen 16 coincides with the main attractor P[—26].

In this way, we can see that the melting points of the elements of
highest abundance in the solar system and the Galaxy (H, 4 He, 16 0)
coincide with main attractors of stability.

Now you can comprehend that the correspondence of the average
temperature of the CMBR with the main attractor P[— 29] cannot as¬
tonish anybody who is familiar with the FF. Within Global Scaling, the
CMBR represents not more and not less than a stable energy level of the
omnipresent protons and electrons. For lack of empirical confirmation,
the Big Bang cosmology overrates the CMBR dramatically.

Actually, any process that corresponds with attractors of proton and
electron stability is of cosmological significance, because it forms that
universe we experience every day. Every atom is a universe, every living
cell is a universe, every animal and each of us is a universe, the Earth
is a universe, the solar system is a universe and the Milky Way is a
universe. All the galaxies together form the universe of the universes
that follows the same Global Scaling law of the number continuum as
do all the other embedded universes.

1 Periodic table of elements. Los Alamos National Lab., www.periodic.lanl.gov

Interscalar Cosmology

The countless worlds in the universe
are no worse and no less inhabited
than our Earth.

Giordano Bruno

Perhaps you are starting to understand that the abundance of coinci¬
dences we mentioned in the introduction of this book is caused by the
FF of space-time that leads to interscalar cosmology, a completely new
understanding of the universe.

Let’s take some more examples. Saturn’s body radius 1 coincides
with the main equipotential surface P[54] of proton stability:

, ( r Saturn \ _ . ( 6.0268 • 10 7 m \

n U proton ) ~ ln V 2.103089 • 10-!6 m) ~ °

The main Theta-wave frequency 5 Hz of brain activity has the same
wavelength and coincides with the same attractor P[—54]:

5 Hz

1.425486 • 10 24 Hz

In this way, the FF creates an abundance of interscalar connections. The
FF connects not only biological frequencies, but also other biophysical
characteristics with astrophysical processes. For example, the average
adult human brain 2 mass 1.4 kg corresponds with the main attractor
P[62] of proton stability (1.6726219 • 10 -27 kg is the proton rest mass):

^ theta
^ proton

^ human brain
^ proton

= In

1.4 kg

1.672622 • 10 ~ 27 kg

= 62

The proton attractor P[62] coincides with the electron attractor E[69;2],
because 69 2 = 62 + 7 5 . Amazingly, the doubled logarithm 69 \ + 69 3
= 139 returns the attractor of electron stability E[139] that is occupied
by the body mass of the Sun:

( M gun \ _ ln f 1.98855 • 10 30 kg \
\ nr electron / ^9.109383 • 1(H 31 k g y

139

1 Saturn Fact Sheet. NASA Space Science Archive, www.nssdc.gsfc.nasa.gov
2 Singh D. et al. Weights of human organs at autopsy. JIAFM, vol. 26 (3), 2004.

Interscalar Cosmology

Therefore, we can write down the equation:

A? Sun \ _2 In ^ m ^ uman

^ electron / \ ^ electron

Applying the FF, we discovered that the mass of the Sun as well as
the average human brain mass correspond with main attractors of sta¬
bility which have an interscalar connection given by the ratio of their
logarithms.

In mathematics, the ratio of logarithms is called fractal dimension
of similarity. First introduced in 1919 by Hausdorff 1 , it is a standard
measure of fractality of both structures and processes.

Before going ahead with mathematics, I would like to mention one
more interscalar connection: Whereas the human brain mass corre¬
sponds with the attractor P [62] of proton stability, the Sun’s surface
gravity (p. 31) corresponds with the same main attractor E[62], but of
electron stability! Therefore, we can write the equation:

^ brain _ ^ electron

^ proton 9 Sun

In this equation, m brain = 1-4 kg is the average adult human brain
mass, m p ro ton = 1.672622- 1CP 2 ' kg is the proton rest mass, a e i ec tron =
2.327421-10 29 m/s 2 is the electron angular acceleration, g Sun = 274 m/s 2
is the Sun’s gravity acceleration (at the photosphere). Rewriting this
equation, we get two equal forces:

^ brain * 9 Sun — ^ proton ‘ & electron

By the way, in this equation proton and electron can be interchanged:

HI electron * ^proton ^-proton * d electron 389.289-J N

So we have got an astonishing result: the human brain in the gravity
held of the Sun weighs exactly the same as the electron weighs in the
acceleration held of the proton!

All by chance? I don’t think so. First, the human brain mass cor¬
responds with a main attractor of proton stability. Second, the Sun’s
gravity acceleration corresponds with the same attractor of electron sta¬
bility. Third, the rest masses of proton and electron are fundamental
constants.

Hausdorff F. Dimension und aufieres Mali. Mathematische Annalen, vol. 122,
issue 1-2, 157-179, 1919.

Interscalar Cosmology

What could be the meaning of this remarkable coincidence? I would
like to propose an astrobiological 1 interpretation. Obviously, biological
organisms are part of the solar system and consequently, their physical
characteristics are embedded in the system.

Let’s see some more examples. As you already know, the Sun’s
radius coincides with the main equipotential surface E[49] of electron
stability. Dividing by 2 gives us the logarithm 49/2 = 24 3 of the
wavelength A electron • exp(24.5) = 16.6 mm that coincides with the focal
length of the human eye that is also the length of the newborn eyeball.

As you know, the radius of Saturn coincides with the main equipo¬
tential surface P[54] of proton stability. Dividing by 2 we receive the
logarithm 27 = 54/2 of the wavelength A pro ton ■ exp(27) = 0.11 mm that
coincides with the size of the human fertile oocyte, the zygote.

Probably, biological organisms on exoplanets are also embedded in
their systems and therefore we can expect that the physical character¬
istics of their physiology correspond with the physical characteristics of
their sun and planets.

To continue this topic let us look at our own organism through the
FF. We begin with the weights of hormonal glands 2 . Figure 8 shows
the correspondence of their weights with the main attractors. As an
example let’s analyze the weight of the hypophysis that statistically is
around 500 mg:

^ hypophysis
^ proton

/ 5 • 10- 4 kg A

V 1.672622- 10- 27 kg J

As you can see, the weight of the hypophysis coincides with the main
attractor P[54] of proton stability. Doesn’t this correlation remind you
of something? Right, the main frequency of Theta brain activity co¬
incides with the same attractor P[—54] of proton stability! The minus
sign isn’t significant — it changes to plus if you switch numerator and
divisor.

In human EEG studies, the term “Theta” refers to frequency com¬
ponents in the 4-7 Hz range, regardless of their source. Indeed, due to
the density of its neural layers, the hippocampus generates some of the
strongest EEG signals of any brain structure, known as the hippocampal
Theta rhythm . 3

1 Muller H. Astrobiological Aspects of Global Scaling. Progress in Physics, issue
1, 3-6, 2018.

2 The GS-analysis of endocrine glands was made by doctor Leili Khosravi.

‘ ? Buzsaki G. Theta Oscillations in the Hippocampus. Neuron, vol. 33, 2002.

Interscalar Cosmology

In vertebrate anatomy, the pituitary gland, or hypophysis, is a pro¬
trusion off the bottom of the hypothalamus at the base of the brain.
One of the most important functions of the hypothalamus is to link the
nervous system to the endocrine system via the pituitary gland.

The hypophysis is often referred to as the “master gland”, because
it controls several of the other hormone glands (adrenals, thyroid) and
many biological functions: metabolism, growth, sexual activity, preg¬
nancy, childbirth, nursing, blood pressure, temperature regulation, pain
relief and stress response.

Do you remember the wavelengths corresponding to the Theta fre¬
quency range? It’s easy to calculate: The main Theta frequency is 5
Hz, and the speed of light is nearly 300,000 km/s. Consequently, the
wavelength is 300,000/5 = 60,000 km. This wavelength is in the scale
of the body radii of Jupiter and Saturn, the largest and most massive
planets, the “master planets” of the solar system. As you remember,
we discovered already that the radius of Saturn coincides with the main
attractor P[54] of proton stability (p. 39).

The coincidence of the Theta rhythm, the mass of the hypophysis
and the radius of Saturn with the same main attractor P[54] of proton
stability demonstrates how the FF connects processes of very different
nature and scales. Interscalar connections are one of the most important
features of the FF.

These connections are caused by the attractors of stability. If the
frequency of some process coincides with an attractor of proton sta¬
bility, then this frequency has a transcendental ratio to the natural
frequency of the proton and consequently, it remains stable because
it avoids proton resonance. Indeed, if the frequencies of two or more
processes coincide with the same attractor of proton stability, the reso¬
nance probability between those processes increases and facilitates the
communication between those processes. In this way, the attractors of
proton and electron stability create stable channels of interscalar com¬
munication between processes which can be of very different scales.

Global Scaling explains the universe in terms of quantum oscillations
and their stability. In the case of quantum oscillators like proton and
electron, their metric characteristics are directly connected with fre¬
quencies through fundamental physical constants — the speed of light
and the Planck constant. Therefore, the ratios of velocities, accelera¬
tions, energies or masses can be attributed to ratios of frequencies and
expressed in terms of resonance and communication probability as well.

Some of the illustrious interscalar connections we already discussed
in this book: between the human brain mass and the mass of the Sun;

Interscalar Cosmology

between the focal length of the human eye and the radius of the photo¬
sphere of the Sun; between the size of the human zygote and the radius
of Saturn (pp. 39-41). Through the FF, biological processes are em¬
bedded in the giant network of interscalar connections in the universe.
Let’s discover some more of them.

The Solar mass coincides with the main attractor E[139] of elec¬
tron stability that coincides with attractor P[131;2] of proton stability,
because 139 — 7 J = 131 Dividing the logarithm 131.5/2 = 65.75
we receive a logarithm that corresponds to the significant sub-attractor
P[ 66 ;—4] in the range of the global average adult human body mass:
rrip ■ exp(65.75) = 60 Kg.

In this way, the human brain mass is connected with the Sun’s mass
through the attractor E[139] of electron stability whereas the human
body mass is connected with the Sun’s mass through the attractor
P[131;2] of proton stability. Consequently, the logarithm of the hu¬
man body-to-brain mass ratio is exactly one half of the logarithm of the
proton-to-electron mass ratio:

2 In I ^ human body \ _ ^ ^ proton

\ ^ human brain / \ ^ electron

This equation you can rewrite in the form:

' \2
^ human body \ _ ^ proton

^ human brain / ^ electron

Jupiter’s body mass corresponds with the main attractor E[132] of elec¬
tron stability:

ln / M Jupiter \ = / 1.8986 • 10 27 kg \

11 \nrelectron/ “ V9.109383 • lO" 31 kg)

132

This attractor coincides with the attractor P[124;2] of proton stability,
because 132 - 7 j = 124 j. The half value of this logarithm 124.5/2
= 62.25 corresponds to m p ■ exp(62.25) = 1.8 kg that is the average
weight of the adult human liver. It is remarkable that the most massive
planet of the solar system corresponds with the most massive organ of
the human organism — the liver:

( m human liver \ J/ Jupiter

nr proton / nr proton

Considering the difference of the logarithms 139 — 132 = 7 of the Sun’s
mass and Jupiter’s mass, you can express the average mass of the human

Interscalar Cosmology

liver also in units of the Sun’s mass. Saturn’s body mass is near the
sub-attractor P[123;4] of proton stability:

( M Saturn \ ( 5.6836 ■ IQ 26 kg \ _ 1

V m proton ) n V 1.672622 • 10-27 kg/ “ 4

The half value of this logarithm 123.25/2 = 61.625 corresponds to
m p -exp(61.625) = 0.97 kg that is the average weight of the adult human
lungs. It is remarkable that the second massive planet of the solar sys¬
tem corresponds with the second massive organ of the human organism
— the lungs:

( TP human lungs \ _ dr Saturn

^ proton / ^ proton

Now you can imagine that aliens who know Global Scaling and have
studied the solar system can predict the human anatomy — at least the
average weight of the human body, of its brain and the main organs
including the glands, the size of the human zygote and the focal length
of the human eye, the frequency ranges of the brain activity, the heart
beat and breathing rates. Indeed, that’s not all they can predict, as we
will see soon.

The average birth weight of a full-term newborn is typically in the
range of 2.5-5 kg 1 . The absolute record is 6.02 kg. These newborn body
weights cover the range between the main attractors E[70] and E[71] of
electron stability:

m = 7i

\ 771 electron /

m (- 2 ^-) = 70

The average birth weight of babies in Europe is 3.6 kilograms and cor¬
responds with the main attractor P [63] of proton stability:

In = 63

\ Tfl proton /

The logarithm 63 is exactly one half of the logarithm of the Venus-
to-electron mass ratio that coincides with the main attractor E[126] of

1 Janssen P. A. et al. Standards for the measurement of birth weight, length
and head circumference at term in neonates of European, Chinese and South Asian
ancestry. Open Medicine, vol. 1 (2), e74-e88, 2007.

Interscalar Cosmology

electron stability:

( M Venus \ _ ( 4.8675 • 10 24 kg

11 \ TO electron / R \9.109383 • 10“ 31 kg J

Consequently, our aliens which know the FF and have studied the solar
system would know also the average human newborn body weight.

The average brain mass of full-term newborns is in the range of 350 g
corresponding with the main attractor E[68] of electron stability:

0.35

kg\

^ electron /

= 68

The average total body length 33-55 cm of full-term newborns is be¬
tween the main attractor P [35] of proton stability and the main attractor
E[28] of electron stability:

0.55 m

^ electron

= 28

0.33 m \

proton

By the way, the normal head circumference for a full-term infant is
33-35 cm at birth.

Now let’s analyze the average adult human body height 1 . Currently
it is in the range of 147 cm (Guatemala, Bangladesh) to 186 cm (Bosnia
and Herzegovina). The shortest adult human was recorded in Nepal at
55 cm. Its noticeable that his body height coincides with the maximum
body length of a newborn. The tallest woman in medical history was
recorded in China, who stood 248 cm when she died at the age of 17.
The tallest living man is recorded in Turkey, at 251 cm:

2.5 m \

proton /

1.5 m

electron

= 29

Obviously, the adult human body height is between the main attrac¬
tor E[29] of electron stability and the main attractor P[37] of proton
stability.

Knil A. J. et al. Self-reported and measured weight, height and body mass
index (BMI) in Italy, the Netherlands and North America. European Journal of
Public Health , vol. 21, 414—419, 2010.

Interscalar Cosmology

If now we put together all ranges of the human body height — from
the newborn minimum to the adult maximum, then it covers the range
between the main attractors P [35] and P [37] of proton stability. The
main attractor P [36] represents the logarithmic mean of this range that
corresponds with the body height of 2.103089 • 10~ 16 • exp(36) = 90 cm
that is typical for children in the age of 2 years. In this age, the baby
becomes a toddler. This development stage is accompanied by a peak
in the brain growth 1 , enormous language improvements, accelerated
learning and self-awareness.

And so, the human body height covers the range P[36±l] of proton
stability. The double logarithm 36 + 36 = 72 corresponds with Jupiter’s
surface gravity that coincides with the main attractor [—72] of proton
stability, as we have seen on page 31. That’s another example of how
biometrics is embedded in the solar system.

It is remarkable that many species develop body sizes and weights
which coincide with main attractors of the FF. In 1981, the biologist
Cislenko 2 discovered that the adults of various species prefer always the
same quite narrow ranges of body sizes. These ranges show an equidis¬
tant logarithmic distribution. Cislenko estimated the scaling factor that
connects one range with the next being close to 3. He analyzed the adult
body sizes of ca. 4700 species of mammals, 5000 species of reptiles, 740
species of fish, 690 species of birds, 21000 species of insects and 900
species of bacteria.

Scale invariance as a property of the metric characteristics of biolog¬
ical organisms is well studied 3 and it is not an exclusive characteristic
of adult physiology. In 1982, Zhirmunski and Kuzmin 4 discovered scal¬
ing in the sequence of the development stages in embryo-, rnorpho- and
ontogenesis and supposed Euler’s number being the scaling factor.

Within the current paradigm in biology, the phenomenon of gener¬
ally preferred body sizes is difficult to explain. Why should it be equally
advantageous for adult fish, amphibians, reptiles, birds and mammals
of thousands of species to have body sizes always in the same ranges?

Cislenko assumed that in the realm of animals and plants there is
not only a competition for food, water or other resources, but also a

HUiickmeyer R. C. et al. A structural MRI study of human brain development
from birth to 2 years. The Journal of Neuroscience, vol. 28(47), 12176—12182, 2008.

2 Cislenko L. L. The Structure of the Fauna and Flora in connection with the
sizes of the organisms. Moscow, 1981.

3 Schmidt-Nielsen K., Scaling. Why is the animal size so important? Cambridge
University Press, 1984.

4 Zhirmunsky A.V., Kuzmin V. I. Critical levels in developmental processes of
biological systems. Moscow, Nauka, 1982.

Interscalar Cosmology

struggle for a favorable body size. Each species tries to occupy an
“advantageous” section on the logarithmic scale, whereby the mutual
competitive pressure creates “crash zones”. However, why both the
“crash zones” and the overpopulated sections on the logarithmic scale
are always of the same width, have the same distance from each other
and why only certain ranges of body sizes are advantageous for the
survival of the species and what these advantages are, could not be
clarified. There are many studies confirming scaling in biology, although
the deep causes have so far remained undiscovered.

Let us take a moment to think about this. The basic idea of Global
Scaling is that the number continuum already contains the solution for
lasting stability in systems of any degree of complexity. Therefore, it
is not necessary to discover this solution by chance through a chaotic
search by competition struggle. The solution is given a priori and it is
omnipresent.

Another examples are the boundaries of the brain activity frequency
ranges Theta, Alpha and Beta, which coincide with main attractors of
the FF and are common for all mammals. We are talking not exclusively
about human physiology, but about biophysics 1 as a whole.

Because of the universality of the FF, Schumann resonances 2 coin¬
cide with attractors which define also the boundaries of brain activity
frequency ranges Theta, Alpha and Beta. This coincidence demon¬
strates that the electromagnetic activity of biological systems is em¬
bedded in the electromagnetic activity of the Earth. Furthermore, this
coincidence allows for interscalar communication, that is, sharing of in¬
formation between processes of different scales.

Let’s analyze the Schumann resonances, starting with the fundamen¬
tal mode of 7.83 Hz:

^ Schumann 1
^ electron

= In

7.83 Hz

7.763441 • 10 20 Hz

-46.04

Variations of the resonance frequencies can be caused by solar X-ray
bursts 3 . In this case, the resonance frequency increases up to 8.2 Hz
reaching the main attractor point E[—46] of electron stability. The

1 Miiller H. Chain Systems of Harmonic Quantum Oscillators as a Fractal Model
of Matter and Global Scaling in Biophysics. Progress in Physics , issue 4, 231-233,
2017.

2 Schumann, W. O. Uber die strahlungslosen Eigenschwingungen einer leitenden
Kugel, die von einer Luftschicht und einer Ionospharenhiille umgeben ist. Zeitschrift
fur Naturforschung A, Bd. 7 (2), 149-154, 1952.

3 Roldugin V. C. et al. Schumann resonance frequency increase during solar
X-ray bursts. Journal of Geophysical Research, vol. 109, A01216, 2014.

Interscalar Cosmology

frequency 14 Hz of the 2 nd mode coincides with the main attractor
P[—53] of proton stability:

^ Schumann 2
^ proton

= In

14 Hz

1.425486 • 10 24 Hz

-52.99

It is remarkable that solar activity affects this mode much less or does
not affect it at all. Indeed, the 3 rd mode frequency 20.3 Hz must increase
up to 22.2 Hz for reaching the main attractor point E[—45] of electron
stability:

^ Schumann 3
^ electron

= In

20.3 Hz

7.763441 • 10 2 ° Hz

-45.09

Schumann resonance modes can reach frequencies up to 60 Hz coinciding
with the main attractor E[—44] of electron stability:

/ ^ Schumann max \ _ In I ^ I

V W electron ) ~ \ 7.763441 • 10 2 ° Hz )

The electromagnetic activity of biological systems is not only embedded
in the electromagnetic activity of the Earth, but also in that of the
Sun. It is notable that interscalar communication is not limited to low
frequencies, but concerns also biophysics of light. For instance, the
boundary between ultraviolet B and C light is close to the wavelength
280 nm that coincides with the main proton attractor P [21]:

. ( Auvb-cA = , ( 2.8 • 10 ~' m A =

11 1 A proton ) 11 V 2.103089-10- 16 mj

The essential for animals aromatic amino acids like tryptophan 1 have a
peak of absorption at the wavelength 280 nm. You remember, this is
also the size of the smallest living cells, the mycoplasma (p. 35).

The boundary between infrared B and C light is close to the wave¬
length 3.4 fim that coincides with the main electron attractor E[16]:

/ Airb-cA .... ( 3.4 • 10~ 6 m A

11 \ a electron/ H \3.861593 • 10 -13 m/

A note on attractor points: As you can see in figure 2 on page 17,
near an attractor point, the FF increases its density and in the point

1 Vashchuk V. et al. Optical Response of the Polynucleotides-Proteins Interac-
tion. Molecular Crystals and Liquid Crystals , vol. 535, 2011, issue 1, 93-110.

Interscalar Cosmology

compression changes to decompression. This inversion has real conse¬
quences.

For example, UV B and C exhibit different properties of their in¬
teraction with atoms or molecules. The same is valid for IR B and C.
In this way, FF-attractors mark not only islands of stability, but also
points of change. Also the boundaries of the brain activity frequency
ranges Theta, Alpha and Beta which coincide with main attractors of
the FF, are points where the brain activity changes.

The change of compression to decompression near attractor points of
the FF can have very different consequences. In geology, main equipo-
tential surfaces of the FF coincide with shells of the Earth’s interior
where seismic waves change their velocity dramatically indicating sharp
density boundaries. 1

The propagation speed of seismic compression waves depends on the
density and elasticity of the medium and therefore we can expect that
they correspond with zones of compression and decompression near the
main equipotential surfaces of the FF.

Figure 9 shows the linear 2D-projection of the FF in the interval
between the main equipotential surface P[49] and the equipotential sub¬
surface P[52;—4] of proton stability. At the graphic’s left side the cor¬
responding radii in km are indicated. The radial distribution of equipo¬
tential surfaces represents the 2D-profile of the Earth’s interior the FF
is suggesting. The minimum and maximum values of the Earth’s radius
approximate the equipotential surface E[44;4] of electron stability:

T Earth max

electron

/ 6.384 • 10 3 m \

V 3.861593- 10- 13 m )

44.252

6.353 -10 3 m \ „ „ _

-—— ) = 44.247

3.861593 • ICR 13 m )

Figure 9 on the next page shows this attractor as dotted line in the top
of the graphic.

Detailed seismic studies have shown that the speed of seismic P-
waves (longitudinal pressure waves) in the mantle increases rather rapid¬
ly from about 9 to 11 km/s at depths between about 400 and 700 km,
marking a layer called the transition zone. This zone separates the
upper mantle from the lower mantle.

1 Muller H. Quantum Gravity Aspects of Global Scaling and the Seismic Profile
of the Earth. Progress in Physics, issue 1, 41—45, 2018.

Earth min

= In

Interscalar Cosmology

In the FF, this transition zone corresponds with the compression
zone before the sub-attractor P[52;—3] at a the distance of 5770 km
from the Earth’s center.

[49] = 400 km

Figure 9: The Fundamental Field with equipotential surfaces of proton sta¬
bility in the linear 2D-projection for k = 1 in the interval [49] ... [52;—4].
Radius in km (left side). The dotted line at the top indicates the Earth surface
that coincides with the equipotential surface E[44;4] = 6372 km of electron
stability.

As they travel more deeply into the mantle, P-waves increase their
speed from 8 km/s at the Mohorovicic discontinuity to about 13 km/s
at a depth of 2900 km. Once P-waves penetrate below 2900 km, their
velocity suddenly drops from 13 km/s back down to about 8 km/s.

This dramatic reduction in speed at the depth of 2900 km defines
the boundary between the Earth’s mantle and the core. The outer core
seems liquid, because seismic S-waves (transversal shear waves) do not
pass this boundary. In contrast, the innermost part of the core within a
radius of 1250 km seems solid. Reaching the inner core, P-waves again
jump to a velocity of 11 km/s. 1

Both standard models PREM 2 and IASP91 3 identify these bound-

1 Ken nett. B. L., Engdahl E. R. Travel times for global earthquake location and
phase identification. Geophysical Journal International , vol. 105, 429—465, 1991.

2 Dziewonski A. M., Anderson D. L. Preliminary reference Earth model. Physics
of the Earth and Planetary Interiors , vol. 25, 297-356, 1981.

3 Kennett B. L. N. IASPEI 1991 Seismological Tables. Canberra, 1991.

Interscalar Cosmology

aries with the radius of the liquid core (3480 km) and the radius of the
inner solid core (1250 km). These estimations correspond with the com¬
pression zones before the main equipotential surfaces P [51] and P[50]
and confirm that P-waves increase their velocity in the compression
zone before the attractor. Then in the decompression zone, after the
attractor, they decrease the velocity.

The pure compression zone of an attractor of proton stability begins
always after the equipotential sub-surface [nO; ±6] that coincides with
the equipotential sub-surface [nO; ±3] of electron stability, because 1/2
- 1/3 = 1/6 (see figure 3 on page 22). The sub-surface P[51;6] of proton
stability returns the radius 3500 km and the sub-surface P[50;6] has
the radius 1290 km. This coincidence is a strong confirmation of the
FF and suggests that the physical characteristics of the Earth’s interior
stratification are not casual, but an essential condition of its stability.

In accordance with the FF, the inner core should also have a sub¬
structure that originates from the equipotential surface P[49] at the
distance of 400 km from the center. The compression zone of this at¬
tractor begins with the distance of P[49;6] = 475 km from the center. In
fact, the seismological exploration of the Earth’s inner core has revealed
unexpected structural complexities. There is a well-defined hemispheri¬
cal dichotomy in anisotropy and also evidence of a subcore with a radius
300-600 km. 1

The FF predicts two additional zones of change which correspond
with the equipotential surfaces P[51;3] of 4150 km radius and P[51;2]
of 4890 km radius. The standard models PREM and IASP91 don’t
mention these peculiarities. Maybe they will be discovered.

Now let’s pay attention to another interscalar connection: Whereas
the radius of the Earth’s subcore corresponds with the equipotential
surface P[49] of proton stability, the radius of the Sun coincides with
the same equipotential surface E[49], but of electron stability. So we
can write down the equation for the ratio of the radii:

T Sun _ ^ electron

t* Earth subcore ^ proton

In this example you can see how the knowledge of the FF allows for the
discovery of interscalar connections which no one could imagine before.
The electron-to-proton ratio we can find many times in the solar system.
Let me give one more example.

1 Deguen R. Structure and dynamics of Earth’s inner core. Earth and Planetary
Science Letters, vol. 333—334, 211—225, 2012.

Interscalar Cosmology

As you already know, Saturn’s body radius coincides with the main
equipotential surface P[54] of proton stability (p. 39). At the same
time, Venus’ mean orbital distance coincides with the same equipotential
surface E[54], but of electron stability (p. 30). Therefore, we can write
down the equation:

A Venus ^ electron

f Saturn ^ proton

Talking about the radius of the Sun or that of a gas giant like Saturn,
there is no solid surface connected with it. The visible diameter of the
Sun is its photosphere whereas the visible diameter of Saturn is the
boundary of its atmosphere or more precisely its stratosphere. This
fact leads to the suggestion that the FF is forming the stratification
of planetary and stellar atmospheres as well. Let’s check this idea and
analyse the stratification of the Earth’s atmosphere.

The vertical stratification of the Earth’s atmosphere is caused by
very different processes and it is a complex field of research. In general,
air pressure and density decrease exponentially with altitude, but tem¬
perature, ionization and chemical composition have more complicated
profiles.

The standard division into troposphere, stratosphere, mesosphere,
thermosphere, ionosphere and exosphere is based on satellite, airplane
and ground measurements and considers aerodynamic, hydrodynamic,
thermodynamic, electromagnetic, chemical and gravitational factors in
their complex interaction.

Stratification as atmospheric feature is associated not only with the
Earth, but occurs on any other planet or moon that has an atmosphere
as well. Furthermore, stable atmospheric boundaries like tropopause,
stratopause, thermopause and mesopause have similar vertical distribu¬
tions at different celestial bodies in atmospheres of very different chem¬
ical compositions.

Being gas, the atmosphere is bounded by the lithosphere and the
hydrosphere of the planet. The lowest layer of Earth’s atmosphere is
the troposphere where nearly all weather conditions take place. The
average height of the troposphere 1 is 20 km in the tropics, 12 km in the
mid latitudes, and 7 km in the polar regions in winter. Table 3 and
figure 10 show the correspondence of these tropospheric levels with the
equipotential surfaces E[37;2], E[38] and E[38;2] of electron stability.

At its lowest part, the planetary boundary layer (PBL), the tropo¬
sphere displays turbulence and strong vertical mixing due to the contact

1 Danielson, Levin, Abrams. Meteorology. McGraw Hill, 2003.

Interscalar Cosmology

with the planetary surface. The top of the PBL 1 in convective condi¬
tions is often well defined by the existence of a stable capping inversion,
into which turbulent motions from beneath are generally unable to pen¬
etrate.

The height of this elevated stable layer is quite variable, but is gen¬
erally below 3 km. Over deserts in mid-summer under strong surface
heating the PBL may rise to 4-5 km. In the temperate zones, it can
be defined by the quite sharp decrease of aerosol concentration at the
height of about 1600 m. Over the open oceans, but also at night over
land, under clear skies and light winds, with a capping stratocumulus,
the depth of the PBL may be no more than 600 m.

Table 3 shows the correspondence of the PBL features with the main
equipotential surfaces E[35], E[36] and E[37] of electron stability. Above
the PBL, where the wind is nearly geostrophic, vertical mixing is less
and the free atmosphere density stratification begins.

The jet stream flows near the boundary between the troposphere
and the stratosphere. As altitude increases, the temperature of the
troposphere generally decreases until the tropopause.

At the bottom of the stratosphere, above the tropopause, the tem¬
perature doesn’t change much, but at the inverse layer at altitudes be¬
tween 20 and 33 km the temperature increases from —50°C to 0°C.
Then at the stratopause at 55 km altitude the temperature stabilizes.
The stratopause is the boundary between two layers: the stratosphere
and the mesosphere 2 .

The ozone layer (ozonosphere) of the stratosphere absorbs most of
the Sun’s ultraviolet radiation and is mainly found at altitudes between
12 and 30 km, with the highest intensity of formation at 20 km height 3 .
Figure 10 shows the correspondence of the main stratosphere layers with
the equipotential surfaces E[39] and E[39;2] of electron stability 4 .

Above the stratopause, in the mesosphere between 55 and 90 km
altitude 5 , the temperature decreases again, reaching about —100°C at

1 Garratt J. R. Review: The atmospheric boundary layer. Earth-Science Review,
vol. 37, 89-134, 1994.

2 Brasseur G. P., Solomon S. Aeronomy of the Middle Atmosphere. Chemistry
and Physics of the Stratosphere and Mesosphere. Springer, 2005.

3 Stolarski R. et al. Measured Trends in Stratospheric Ozone. Science, New
Series, vol. 256, issue 5055, 342-349, 1992.

4 Mliller H. Global Scaling of Planetary Atmospheres. Progress in Physics, issue
2, 66-70, 2018.

’Holton J. R. The Dynamic Meteorology of the Stratosphere and Mesosphere.
Meteorological monograph, vol. 15, no. 37, American Meteorological Society, Boston
(Massachussetts), 1975.

Interscalar Cosmology

m oooei.

m 0028

m ooos

m oooe

m 0081.

m oou

w>| 099

LU>I OOP

lu>) 0S2

UJ>| OSL

UJ>| 06

m ss

lu>i ce

wyoz

un\Zl

w* S l

w* 9P

lu>| 9' l

lu>I 9 0

Interscalar Cosmology

BOUNDARY

ALTITUDE h, KM

In (h/X e )

van Allen outer electron belt

13000

44.96

E[45]

8200

E[44;2]

5000

E[44]

van Allen inner proton belt

3000

43.50

E[43;2]

Earth exopause

1800

42.99

E[43]

1100

E[42;2]

Earth thermopause

650

41.97

E[42]

400

E[41;2]

Venus & Mars thermopause
Venus atmospheric entry

250

41.01

E[41]

Earth atmospheric entry

Venus mesopause

150

40.50

E[40;2]

Earth & Titan mesopause
Venus tropopause

Mars stratopause & entry

39.99

E[40]

Earth & Titan stratopause

39.50

E[39;2]

Titan tropopause

38.99

E[39]

Earth tropic tropopause

38.49

E[38;2]

Earth temperate tropopause

37.98

E[38]

Earth polar tropopause

7.5

37.51

E[37;2]

desert summer PBL inversion

4.5

37.00

E[37]

continental PBL inversion

1.6

35.96

E[36]

marine PBL inversion

0.6

34.98

E[35]

Table 3: The atmospheric stratification boundaries on Earth, Venus, Mars
and Titan and their correspondence with equipotential surfaces of electron
stability.

Interscalar Cosmology

the mesopause 1 . The mesopause corresponds with the main equipoten-
tial surface E[40] of electron stability. This altitude of 90 km coincides
with the turbopause: above this level the atmosphere is of extremely
low density so that the chemical composition is not mixed but stratified
and depends on the molecular masses.

Above the mesopause, in the thermosphere, the (kinetic) temper¬
ature increases and can rise to 1000°C (depending on solar activity)
at altitudes of 250 km remaining quasi stable with increasing height.
Due to solar radiation, gas molecules dissociate into atoms: above 90
km carbon dioxygen and dihydrogen dissociate, above 150 km dioxy¬
gen dissociates and above 250 km dinitrogen dissociates. Above 150
km, the density is so low that molecular interactions are too infrequent
to permit the transmission of sound. These thermospheric layers corre¬
spond with the main equipotential surfaces E[40;2] and E[41] of electron
stability.

The Karman line 2 is considered by the Federation Aeronautique In¬
ternationale (FAI) 3 as the border between the atmosphere and outer
space, as altitude where the atmosphere becomes too thin to support
aeronautical flight, since a vehicle at this altitude would have to travel
faster than orbital velocity to derive sufficient aerodynamic lift to sup¬
port itself.

On Earth, atmospheric effects become noticeable during atmospheric
entry of spacecraft already at an altitude of around 120-150 km, while
on Venus the atmospheric entry occurs at 250 km and on Mars at about
80-90 km above the surface. These heights mark also the bases of the
anacoustic zones where the density of the atmosphere is too low for
sound propagation.

The location of the thermopause is near altitudes of 600 700 km

and depends on solar activity 4 . The thermopause corresponds with the
main equipotential surface E[42] of electron stability. Above starts the
exosphere, where the atmosphere (mostly consisting of hydrogen atoms)
thins out and merges with interplanetary space. This uppermost layer,
until 13,000 km observable from space as the geocorona, extends up to
100,000 km.

1 Beig G., Keckhut P. Lowe R. P. et al. Review of mesospheric temperature
trends. Rev. Geophys., vol. 41(4), 1015, 2003.

2 Karman T., Edson L. The Wind and Beyond. Little, Brown, Boston, 1967.

3 Cordoba S. F. The 100 km Boundary for Astronautics. Federation Aeronautique
Internationale, 2011.

4 Beig G., Scheer J., Mlynczak M. G., Keckhut P. Overview of the temperature
response in the mesosphere and lower thermosphere to solar activity. Reviews of
Geophysics, vol. 46, RG3002, July 2008.

Interscalar Cosmology

The van Allen 1 radiation belts are features of Earth’s magneto¬
sphere. The inner belt consists of high energetic protons which reach
their maximum concentration at altitudes of 3,000 km. The outer belt
consists of high energetic electrons with maximum concentration at al¬
titudes of 13,000 km. The outer belt maximum corresponds well with
the main equipotential surface E[45] of electron stability, but the inner
belt maximum corresponds with the equipotential surface E[43;2] that
is the main equipotential surface P [51] of proton stability. In this way,
the FF delivers not only a correct estimation of their altitudes, but also
a simple explanation of the separation in two belts of high proton and
high electron concentration respectively.

Probably, in future the FF can be applied for estimation of the
atmospheric stratification at ice giants like Uranus and Neptune and
gas giants like Jupiter, Saturn and extrasolar planets as well. Vacant
attractors in table 3 could be identified as stratification features.

Having analysed the solar system, now we venture into more distant
regions of the Milky Way (MW), our home Galaxy. At the same time, we
have to consider that distance measurement by parallax triangulation
is precise enough only up to 500 light years. With the increase of the
distances, indirect methods are applied blurring the difference between
facts and model claims.

Furthermore, all 300 or more billions of stars in the Galaxy are
moving and changing their relative positions and distances continuously.
Nevertheless, analysing the distances between stars with the help of the
FF we can estimate the probability that a distance is currently stable
or intensely changing. In this way, we can also get an idea about the
hierarchy of the stars in a group. Of course, our estimation will be very
hypothetical while it is not possible to verify by observation. Let’s start
with our star neighbourhood. The distance to the Sirius system 8.60(4)
light years fits with the main attractor P[75] of proton stability,

, f R Sirius *\ _i /8.6 • 0.946053 • 10 16 m\

11 \ A proton / n V 2.103089 • 10-!6 m )

whereas the distance to the Alpha Centauri system 4.34(3) light years
coincides with the sub-attractor P [74;3]:

. / i?Alpha Cen\ . / 4.3 • 0.946053 • 10 16 m\ 1

V A proton ) n V 2.103089 • 10 -16 m j + 3

1 Schaefer H. J. Radiation Dosage in Flight through the Van Allen Belt. Aero¬
space Medicine , vol. 30, no. 9, 1959.

Interscalar Cosmology

Knowing the sky coordinates of both Alpha Centauri and Sirius, it
is possible to calculate the distance between them by triangulation 1 .
However, in this book we will not. In general, single stars don’t orbit
each other if they are not members of the same system. However, we
can expect that in general, stars orbit the Galactic Center (GC).

Currently there is no precise measurement of the distance to the
Galactic Center, but 26,000 light years seems an accepted estimation 2
and it coincides with the main attractor P[83] of proton stability:

/I?Sun-Gc\ _ ln /2.6-10 4 - 0.946053 • 10 16 m\
V A p roton ) - 111 ^ 2.103089 • 10- 16 m )

83.05

If the current measurement is correct, it would mean that the solar
system orbits the Galactic Center at a distance that avoids resonance
interaction with it. Good for us. By the way, 26,000 years reminds us
of the precession period of the Earth that coincides with the same, but
temporal attractor P[83] of proton stability:

-I " 1 Earth precession
proton

= In

2.6 • 10 4 • 31558149.54 s
7.015150 • 10- 25 s

83.05

This coincidence of the precession period of the Earth with its distance
to the Galactic Center isn’t random, but indicates a profound connection
of both processes given by the FF. Both processes are of the same scale,
only in one case it is a spatial scale and in the other case it is a temporal
scale and so they meet the same attractor.

Now let’s look beyond our Galaxy at some members of the local
group. It is easy to observe them even with a good binocular, but it
is hard to measure the distances to them. They are too large for di¬
rect parallax triangulation. For determination of intergalactic distances,
Cepheid variable stars in other galaxies are observed.

In 1908 Henrietta Swan Leavitt 3 who was looking for Cepheid stars
in the Magellanic Clouds, discovered a period-luminosity relation for
Cepheids. She found that Cepheids of a high brightness have larger
pulsation periods than those of lower brightness. Thanks to this dis¬
covery, one can measure how often the Cepheid changes luminosity and
calculate its intrinsic luminosity.

1 Hirshfeld A. W. Parallax: The Race to Measure the Cosmos. Dover Publ.,
2002 .

2 Groom D. E. et al. Astrophysical constants. European Physical Journal C.
vol. 15, 1, 2000, www.pdg.lbl.gov

3 Leavitt H. S. 1777 variables in the Magellanic Clouds. Annals of Harvard
College Observatory , vol. 60, 87, 1908.

Interscalar Cosmology

It is believed that for any star, its apparent luminosity (how bright
it appears to us) decreases with the square of the distance to the ob¬
server. If now we measure the apparent luminosity of a Cepheid and
we know its intrinsic luminosity by measuring its period, we can obtain
the approximate distance to the object.

Indeed, some assumptions 1 need to be made before measuring dis¬
tances using Cepheid stars: the period-luminosity relation of all Cepli-
eids must be the same; all Cepheids of one galaxy must be equidistant
from the Earth; their light must not be absorbed by dust clouds.

Considering the uncertainty of these assumptions, we should be care¬
ful with far-reaching interpretations of those measurements. However,
for exercise let us consider them as trustable. Today the distance to the
Large Magellanic Cloud (LMC) is estimated to be 186 thousand light
years and it coincides with the attractor E[77;2] of electron stability:

/i? MW -LMc\ _ lu /1-86 • 10 5 • 0.946053 • 10 16 m\
V A electron / V 3.861593 • 10~ 13 m )

77.50

The distance to the Small Magellanic Cloud (SMC) is estimated with
157 thousand light years and it coincides with the attractor E[77;3] of
electron stability:

In ( ^ MW-SMC \ _ /1.57 • 10 5 • 0.946053 • 10 16 m\ ^ oo

V A electron / “ \ 3.861593 • 10" 13 m )~ '

The Andromeda galaxy M31 seems to be at a distance of 2.5 million
light years 2 that is in the deceleration zone after the last sub-attractor
E[80;6] of both proton and electron stability, very close to the main
attractor E[80] of electron stability:

MW-M31

electron

/ 2.5 • 10 6 • 0.946053 • 10 16 m\
V 3.861593 • 10^ 13 m J

80.10

For reaching the attractor E[80], the Andromeda-to-Milky Way distance
has to decrease by 240 thousand light years down to 2.26 million light
years:

3.861593 • 10” 13 m • exp(80) = 2.26 • 10 6 ly

They seem to do exactly this. M31 is approaching (more precisely,
2.5 million years ago was approaching) the Milky Way at about 100

1 Casertano S. et al. Parallax of Galactic Cepheids. arXiv:1512.09371v2, 2016.

2 Ribas I. et al. First Determination of the Distance and Fundamental Properties
of an Eclipsing Binary in The Andromeda Galaxy. arXiv:astro-ph/0511045vl, 2005.

Interscalar Cosmology

kilometers per second, as indicated by blueshift measurements 1 . If the
velocity of approach is constant, the current distance to M31 should be
already 1,000 light years shorter than the 2.5 million years old distance
we can measure today.

Standard model calculations (naturally without consideration of the
FF) expect that both galaxies will collide in a few billion years. Consid¬
ering the fractality of the FF, we can expect that the approach velocity
is slowly decreasing and after reaching the attractor E[80], the approach
will be finished and the distance between both galaxies will be stabilized
at 2.26 million light years. As you can see, the consideration of the FF
can modify predictions completely.

Talking about galaxies, we can’t avoid mentioning the hypothesis
about dark matter. It is important for us to understand where this
hypothesis is coming from and how it is related with the evidence of the
FF. Hence, let’s spend a few minutes on this topic.

Already in 1933, Fritz Zwicky 2 studied the Coma Cluster and ob¬
tained evidence of unseen mass that he called “dark matter”. In 1957,
Henk van de Hulst and then in 1959, Louise Volders demonstrated that
the galaxies M31 and M33 do not spin as expected in accordance with
Kepler’s laws.

The orbital velocities of stars should decrease in an inverse square
root relationship with the distance from the Galactic Center, similar
to the orbital velocities of planets in the solar system. But this is not
observed. Outside of the central galactic bulge the orbital velocities are
nearly constant.

According to Newton’s hypothesis of mass as source of gravity, this
deviation might be explained by the existence of a substantial amount
of matter flooding the galaxy that is not emitting light and interacts
barely with ordinary matter and therefore it is not observed.

Here it is important to realize that dark matter is required only if
mass causes gravitational interaction. Indeed, exactly this point is still
under discussion.

The origin of gravity is a key topic in modern physics. Furthermore,
gravity is the only interaction that is not described yet by a consistent
quantum theory. The universality of gravity means that the free fall
acceleration of a test body at a given location does not depend on its
mass, physical state or chemical composition.

1 Cowen R. Andromeda on collision course with the Milky Way. Nature.com, 31
May 2012.

2 Zwicky F. On the Masses of Nebulae and of Clusters of Nebulae. The Astro-
physical Journal, vol. 86, 217, 1937.

Interscalar Cosmology

This discovery, made four centuries ago by Galileo Galilei, is con¬
firmed by modern measurements with an accuracy of 10 lo -10 12 . A
century ago Einstein supposed that gravity is indistinguishable from,
and in fact the same thing as, acceleration. In fact, Earth’s surface
gravity acceleration g can be derived from the orbital elements of any
satellite, also from the Moon’s orbit:

H = Ait—r = 3.9860044 • 10 14 m 3 /s 2

(6372000 m) 2

9.82 m/s 2

R is the semi-major axis of the Moon’s orbit, T is the orbital period
of the Moon and r is the average radius of the Earth, g is called the
geocentric gravitational constant. As you can see, no data about the
mass or chemical composition of the Earth or the Moon is needed.

Kepler’s 3 rd law describes the ratio R 3 /T 2 as constant for a given
orbital system. Kepler’s discovery is confirmed by high accuracy radar
and laser ranging of the movement of artificial satellites. Kepler’s 3 rd
law is of geometric origin and can be derived from Gauss’s flux theorem
in 3D-space. The law applies to all conservative fields which decrease
with the square of the distance 1 and does not require the presence of
mass.

Newton’s law of universal gravitation postulates the identity /./ =
GM , an interpretation that provides the mass M as source of grav¬
ity and the universality 2 of the big G. Both postulates are essential
in Newton’s theory of gravitation and in Einstein’s general theory of
relativity.

And yet, they are not essential for precise description and predic¬
tion of the orbital movements in the solar system. Therefore, Newton’s
hypothesis about mass as source of gravity could turn out to be a dis¬
pensable assumption.

In the case of mass as source of gravity, in accordance with New¬
ton’s shell theorem, a solid body with a spherically symmetric mass
distribution should attract particles outside it as if its total mass were
concentrated at its center. In contrast, the attraction exerted on a par¬
ticle should decrease as the particle goes deeper into the body and it
should become zero at the body’s center.

1 Wess J. Theoretische Mechanik. Springer, 2009.

2 Quinn T., Speake C. The Newtonian constant of gravitation — a constant too
difficult to measure? An introduction. Phil. Trans. Royal Society A, vol. 372,
20140253.

Interscalar Cosmology

A boat at the latitude 86.71 and longitude 61.29 on the surface of
the Arctic Ocean, would be at the location that is regarded as having
the highest gravitational acceleration on Earth. At that location, the
gravitational acceleration is 9.8337 m/s 2 . At higher or lower position
to the center of the Earth, gravity should be of less intensity. This con¬
clusion seems correct, if only mass is the source of gravity acceleration
and if the big G is universal under any conditions and in all scales.

The Preliminary Reference Earth Model 1 affirms the decrease of the
gravity acceleration with depth. However, also this hypothesis is still
under discussion.

In 1981 1986, Stacey 2 , Tuck, Holding, Maher and Morris reported 3
anomalous measurements (larger values than expected) of the gravity
acceleration in deep mines and boreholes. Frank Stacey writes: “Mod¬
ern geophysical measurements indicate a 1% difference between values
at 10 cm and 1 km (depth). If confirmed this observation will open up
a new range of physics” . 4

Is it this new range of physics that you touch upon while reading
this book? Like already many times in history, new physics is coming
from mathematics developed centuries ago, but applied only today.

Dziewon.sk i A. M., Anderson D. L. Preliminary reference Earth model. Physics
of the Earth and Planetary Interiors, vol. 25, 297-356, 1981.

2 Stacey F. D. et al. Constraint on the planetary scale value of the Newtonian
gravitational constant from the gravity profile within a mine. Phys. Rev. D, vol. 23,
1683, 1981.

4 Holding S. C., Stacey F. D., Tuck G. J. Gravity in mines. An investigation of
Newton’s law. Phys. Rev. D, vol. 33, 3487, 1986.

4 Stacey F. D. Gravity. Science Progress, vol. 69, no. 273, 1-17, 1984.

Nothing is Artificial in the Universe

Every production, of whatever kind,
is an alteration, but the substance
remains always the same.

Giordano Bruno

The basic idea of Global Scaling is that the solution for lasting stability
in systems of any degree of complexity is an inherent feature of the
number continuum given by the natural exponential function e x for
rational exponents. The solution is given a priori and it is omnipresent.

Naturally, this solution is available for technical systems too. There¬
fore, Global Scaling is significant in engineering as well. Analyzing a
few striking examples, we will see that technology is sensitive to FF-
attractors of stability.

Let’s start with computer technology. One of the significant charac¬
teristics of microprocessors is the clock rate. Analyzing the development
history of microprocessors, we can see that the most popular clock rates
occupy main attractors of stability.

Figure 11 shows the distribution of the clock rates in MHz. In the
top of the graphic you can see the frequency ranges applied to various
generations of Intel processors 1 .

A clock generator is an electronic oscillator circuit that uses the me¬
chanical resonance of a vibrating crystal of piezoelectric material (quartz
or ceramic) to create an electrical signal with a stable frequency. Man¬
ufacturers have difficulty producing crystals thin enough to generate
fundamental frequencies over 30 MHz, so that high frequency crystals
are often designed to operate at third, fifth, or seventh overtones. Please
note that they are not integer multiples of the fundamental frequency.

FF-attractor frequencies are applied not only as clock rates in PCs,
but also in USB-technology (6 MHz), GPS, DECT (10 MHz), 3G, EGA
(16 MHz), VGA, GSM, UMTS (25 MHz), remote controlled cars and
boats (40 MHz), Ethernet (50 MHz), PCI (66 MHz) and as carriers in
FM radio (100 MHz), radio control (333 MHz), cell phone (900, 1400
MHz) and Wi-Fi (2.4 GHz). 2

1 Intel Microprocessor Quick Reference Guide — Product Family, www.intel.com

2 Crystal oscillator frequencies, www.en.wikipedia.org

Nothing is Artificial in the Universe

The transition from frequencies to wavelengths of the proton or elec¬
tron is given by the (constant) speed of light. Therefore, the logarithm
changes only the sign, so that the frequency range between the proton
attractors P[— 37] = 120 MHz and P[—35] = 900 MHz corresponds to
the range of wavelengths between P[35] = 33 cm and P[37] = 250 cm
coinciding with the human body size range.

On the example of crystal oscillators, we have seen that FF-attractor
islands of stability concern mechanical oscillations as well. Always when
high precision stability is required, the preferred physical quantities in
mechanical applications correspond with FF-attractors.

A striking example are the calibers of sportive guns, because of the
advanced requirements to the precision and ballistic stability, and espe¬
cially those of pneumatic construction, because of the applied modest
initial impulse.

The most distributed caliber of sporting air rifles 1 is 4.5 mm (diam¬
eter), for use in international target shooting competition at 10 m, up
to Olympic level in both rifle and pistol events. This type of riffles has
helical grooves called rifling machined into the bore wall. When shoot¬
ing, a rifled bore imparts spin to the projectile about its longitudinal
axis, which gyroscopically stabilizes the projectile’s flight.

Analyzing the rotation radius 4.5 mm/2 = 2.25 mm, we discover
that it fits perfectly with the main attractor P[30] of proton stability:

2.25 • 10~ 3

A proton

30.00

The radius 3.8 mm of the prominent caliber 7.6 mm coincides with the
main attractor E[23] of electron stability:

3.8 • 10" 3 m

A electron

23.01

The proton Compton wavelengths equal: A pro ton = 2.103089 • 10 -16 m
and Aeiectron = 3.861593 • 10 -13 m (see table 1 on page 20).

I have no intention of going any deeper into the study of weapons.
Naturally, it is more a question of ethics than of technology to find out
how to prevent the misuse of technological advances against life. Yet, I
do not wish to support any research in this field.

Instead, let’s analyze some features of modern car technology. One
of the most loaded part of the car construction and exposed to high wear

TToff A. Air-guns and Other Pneumatic Arms. Barrie & Jenkins, London, 1972.

Nothing is Artificial in the Universe

is the wheel. Quietness under high speed rotation and mechanical stress
requires a high level of stability avoiding internal resonance. Therefore,
we can expect that significant physical characteristics of the most dis¬
tributed constructions correspond with FF-attractors of stability. In
fact, the most popular wheel size 1 in the USA and Europe is the R15.
It is installed on 50% of passenger cars and light SUVs. Considering a
fitting tire like 185/80-R15 or 205/75-R15, we always get a total wheel
radius of about 34 cm that corresponds with the main attractor P[35]
of proton stability:

Aproton • exp(35) = 0.33 m

As you can see, the unloaded dimension of the car tire is a little bit
larger than the attractor point wavelength, so that the attractor point
can be reached during the rotation under load. The requirements to
aircraft tires 2 are even higher because of high acceleration by landing
and high load capacity. For example, the B737 main gear tires are of
the type 1144.5x16.5-21, where 44.5 inches is the total diameter of the
tire. Consequently, the radius of the tire is 22.25 inches = 56.5 cm that
corresponds with the main attractor E[28] of electron stability:

Aelectron • exp(28) = 0.56 m

Another highly loaded component is the internal combustion engine.
That’s why we are going to analyze some of its functionally significant
physical characteristics. Lasting stability of the movement of the pistons
with minimum friction losses under conditions of high temperature and
high pressure requires high precision of manufacturing.

Furthermore, quietness is required, resonance vibrations are unde¬
sirable, so we can expect that the functional physical characteristics
should be sensitive to attractors of proton or electron stability.

Regardless of any advanced electronic control, the undisputed law of
combustion engines says: “There’s no replacement for displacement.” In
fact, the engine displacement is functionally highly significant. Analyz¬
ing some widespread displacement volumes, we can see that obviously,
already the volume by itself as physical quantity is sensible to attrac¬
tors of proton or electron stability. For example, the famous 1.9-liter
displacement coincides with the main attractor P[102],

1.86 • lCT 3 m 3

1 The Most Popular Tire Sizes: R15. Capitol Tires. 2018.

2 Aircraft Tire Dimensions: www.boeing.com; Global Aviation Tires:

www.goodyearaviation.com; Aircraft Tire Engineering Data: www.michelinair.com

Nothing is Artificial in the Universe

In fact, the physical displacement volume is a little bit less than 1.9
liters. The fundamental proton unit for volumes is A 3 roton = 9.3019276 •
10~ 48 m 3 (see table 1 on page 20).

Based on the prominent displacement volume 250 cm 3 all scooters 1
are divided in two classes — mini and maxi. You remember that FF-
attractors are also points of change. This boundary volume coincides
with the main attractor P [100]:

2.5 • 10~ 4 m 3
A 3

/ 'proton

Another significant characteristic is the speed of revolution that is for
modern internal combustion (Diesel) engines between 800 (in neutral)
and 6000 revolutions per minute. These limits correspond with the main
attractors P[— 53] respectively P[—51]:

(6000/60) Hz

^proton

(800/60) Hz

^proton

The angular frequency of the proton is w pro ton = 1.425486 • 10 24 Hz.
Consequently, the logarithmic mean rotation speed of 2200/min coin¬
cides with the attractor P[—52] of stability:

(2200/60) Hz

^proton

It is a small step from transport technology to traffic where the driving
speed is a highly significant metric characteristic.

As fundamental characteristic of space-time, the speed of light is a
common property of both proton and electron connecting their natural
frequencies with the wavelengths. Consequently, the attractors of ve¬
locities are the same for proton and electron stability, so we use square
brackets without E or P.

It is remarkable that also traffic tries to avoid resonance interaction
and consequently, traffic is sensitive to FF-attractors of stability. For
example, in many countries the traffic speed on highways is limited,
mostly to 120 km/h that is also the average speed on highways in Ger¬
many where there is no speed limitation. 2 Naturally, the existence of a

1 Scooter (motorcycle), www.en.wikipedia.org

2 Kellermann G. Geschwindigkeitsverhalten im Autobahnnetz 1992. Strasse und,
Autobahn, issue 5, 1995.

Nothing is Artificial in the Universe

limit pushes the drivers to go up to the limit, whenever it is possible.
Therefore, it is important that the limit coincides with an attractor of
stability. In the case of 120 km/h it really does:

( W3.6W S ) = _ 16

The speed of light in vacuum is c = 299, 792,458 m/s. For safety rea¬
sons, the maximum speed of tuned cars registered for public transport
and also of ultralight aviation is limited to 330 km/h that is a main
attractor and consequently, a boundary speed as well:

ln ^(330/3.6) m/s ^

= —15

Statistically, the speed at high traffic levels on highways fluctuates
around the average of 75 km/h. Trying to avoid collision, drivers em¬
pirically find this attractor of stability:

h ((7 5 /3.6)m/.) = _ 16 _l

Interestingly, the line of cars does not stand in the traffic jam, but
moves backwards it gets longer. The jam snake grows in the direction
opposite the traffic flow 1 at an average of 15 kilometers per hour that
coincides with the main attractor [—18]:

In ( (1 5A6>-/» ) = _ 18

The adaption to a main attractor of stability is very understandable if
we consider that physical resonance interaction in a traffic jam would
provoke a disaster.

Here we can begin to see technology in general as not something
artificially created by humans, but as a cosmic phenomenon. Proba¬
bly, many other civilizations in the Galaxy create technology as well.
However, all this technology is part of the universe, it isn’t something
unnatural, it consists of the same natural atoms and it is created by the
universe itself we are only the hands of the universe.

From the point of view of Global Scaling, there is nothing “artificial”
in the universe. Everything, whether man-made or naturally grown,

1 Verkehrsfluss und Stauaufkommen. Definitionen. Bundesamt fiir Strassen, AS¬
TRA. Schweizerische Eidgenossenschaft. www.astra.admin.ch

Nothing is Artificial in the Universe

must take into account the FF, because it defines the distribution of
stability attractors in all scales of the universe.

As examples, let us remember also the seismic waves (p. 51) which
change their velocities from 13 km/s = [—10] downto 8 krn/s = [—10;—2]
on the boundary between the Earth’s mantle and the core. By the way,
Jupiter’s orbital velocity fits with the same main attractor [—10].

Now let’s analyze some other boundaries, for example the world
records in Athletics 1 . In fact, our organism is very responsive to the FF-
attractors and some of these boundaries are insurmountable to anybody.

The world record over 10 km walk is 37 minutes 11 seconds, hold by
the Russian athlete Roman Rasskazov in 2000:

ln( l°°°°.f231s ) = _ 1802

To reach the attractor point [—18], the future world record athlete must
walk the 10 km in 2190 seconds = 36 minutes 30 seconds:

10 km

c • exp(—18) = 4.56 m/s =

By the way, this world record walking velocity coincides with the same
attractor [—18] we mentioned already in the case of growing traffic jam!
The average human walking speed at crosswalks is about 6 km/h =
1.6 m/s. Many people prefer to walk at this speed. Being close to an
attractor of stability, this circumstance appears to be natural:

^1.6 m/s^

-19.05

The world record 9.58 seconds over 100 m running was held by the
Jamaican athlete Usain Bolt in 2009. Here we can see that his running
speed is quite close to the main attractor [—17]:

^ 100 m/9.58 s^j

-17.17

Knowing the FF we can affirm that with high probability, nobody will
be able to exceed the main attractor [—17] and run over 100 m faster
than in 8 seconds:

100 m

c • exp(—17) = 12.41 m/s = ——

1 List of world records in athletics, www.en.wikipedia.org

Nothing is Artificial in the Universe

The world record 2.45 m in high jump was held by the Cuban athlete
Javier Sotomayor:

2.45

proton,

36.99

This record has been held since 1993. This might demonstrate how
unattainable it is. In fact, it is only one centimeter before the main
proton attractor point P[37]:

Aproton • exp(37) = 2.46 m

The attractor P[37] defines a main equipotential surface of the Fun¬
damental Field that affects all processes as well as the growth of the
organism. You remember that P[37] defines also the statistical bound¬
ary for modern human body height.

Let’s analyze also the metric characteristics of some animals. For
example, the Peregrine falcon and the Golden eagle can reach flight
speeds of 320 km/h close to the main attractor [—15] we know already
from the speed limit for ultralight aviation.

Black marlins can swim over large distances with a speed of about
120 km/h. You remember this [—16] attractor speed from the traffic
speed limit on European highways.

Greyhounds are the fastest dogs, and have primarily been bred for
coursing game and racing. They can hold a running speed close to
74 km/h that coincides with the attractor [—16;—2]. You remember
that the speed at high traffic levels on highways fluctuates around this
attractor.

The consideration of biophysical characteristics and limitations of
the human or animal organism is an important topic in civil engineering
and is directly connected with ergonomics. Another not less important
topic is the stability of constructions.

Therefore, civil engineering is another held where the knowledge of
the FF could be useful. The avoidance of resonance under periodic
load in general and seismic stability in particular is and was always
an important topic in civil engineering, especially in the Mediterranean
region and other areas of permanent seismic activity. Therefore, we can
expect that especially large-scale constructions should be responsive to
FF-attractors of stability.

Monolithic domes are instructive examples of very high stability.
They meet FEMA 1 standards for providing near-absolute protection

1 Building Codes. FEMA. www.fema.gov

Nothing is Artificial in the Universe

and have a proven ability to survive tornadoes, hurricanes and earth¬
quakes.

In 1991, twenty-eight monolithic domes all of the same size were built
in Iraq. Twenty-seven of the domes were grain storages. In addition to
these, one more dome was built as a government building in Baghdad.
This construction survived a direct hit by a 2300 kg bomb in 2003. The
interior of the structure was totally destroyed, but the monolithic dome
(inner diameter 117 feet = 35.66 m) itself remained standing except a
hole in the top of the dome. 1

Beginning in 1970, monolithic domes have been built and are in
use in virtually every American state and in Canada, Mexico, South
America, Europe, Asia, Africa and Australia. Coinciding with the main
attractor P[39] diameters of about 36 m are preferred not only in modern
constructions, but were favoured in civilizations in antiquity as well.
Here are some famous examples of dome constructions from classical
antiquity. 2

The dome of the Pantheon 3 in Rome, an unreinforced monolithic
concrete construction with a diameter of 43.7 m and with about 5000
tons of weight, is the archetype of the domes built in the following
centuries both in Christian churches and in Muslim mosques.

The 16 massive Corinthian columns supporting the portico weigh 60
tons each. They are 11.8 m = E[31] tall, 1.5 m = E[29] in diameter and
brought all the way from Egypt. The hole (oculus) in the top of the
dome, 4 m = E[30] in radius, is the only source of light.

The interior space of the Pantheon is completely inside a spheroid
with a radius of 21.85 m that is touching the electron stability sub¬
attractor E[32;—3]:

^electron

• exp

21.85 m

The dome of the St. Peter’s in Rome has a radius of 21.5 m that touches
the proton stability sub-attractor P[39;6]:

^proton * Gxp ^39 T — 21.5m

Both radii are in the compression zone of the main attractor P[39] of

1 Garrison K. Monolithic Mosque in Iraq Still Stands, www.monolithic.org
2 Como M. Statics of Historic Masonry Constructions. Springer, 2013.

3 Cinti S. et al. Pantheon. Storia e Futuro / History and Future, Roma, Gangemi
Editore, 2007.

Nothing is Artificial in the Universe

proton stability:

^proton * exp(39) — 18.2 m

The deviation of the sub-attractors of proton and electron stability is
hardly visible in figure 12 because of the logarithmic representation (in
the scale of this book the logarithmic deviation of 0.015 appears to be
less than 1 mm).

The dome of the Hagia Sophia 1 in Istanbul is of 31.24 m in diame¬
ter, so that the radius is touching the sub-attractor E[31;3] of electron
stability:

^electron

• exp

15.65 m

The Hagia Sophia has survived a big fire in 859 and an earthquake in
869. The dome has collapsed after an earthquake in 989. Due to the
earthquakes in 1344 and 1346 a part of the dome and parts of the arch
have collapsed and have been repaired.

In Piedmont, Francesco Gallo designed for the drum of the Sanctuary
of Vicoforte 2 built by Vitozzi in 1596, one of the largest and complex
elliptical domes ever built, with the large diameter being 36 m. As
you can see in figure 12, the radii of the Hagia Sophia dome and the
Pantheon dome occupy mirror positions in relation to the main attractor
P[39] held by the dome of the Sanctuary of Vicoforte.

What do you think: Considering the explicit nonlinearity of the FF
and the very sophisticated measurements of proton and electron, could
you imagine that this high concurrence of the physical characteristics of
antique constructions with the FF happened by chance? Or should we
become familiar with the idea that Global Scaling is a rediscovery of a
very antique knowledge?

But how and where was this ancient knowledge conserved? May be,
it is well exposed in architecture, but we don’t see it without Global
Scaling glasses?

The German term “Mafiwerk” (tracery) brought me on a trail. A
tracery consists of geometric patterns and is an important element of
Gothic architecture, well exposed in window roses. The term “Mafi¬
werk” derives from “Mafi” (measure) and underlines that the geometric
patterns contain some metric information. In fact, the diameter of the

1 Curcic S. Architecture in the Balkans. From Diocletian to Suleyman the Mag¬
nificent. Yale University Press, New Haven und London 2010.

2 Bagliani S. The Architecture and Mechanics of Elliptical Domes. Proceedings
of the Third International Congress on Construction History, Cottbus, May 2009.

Nothing is Artificial in the Universe

rose window of the Cathedral of Our Lady of Strasbourg is 13.5 nr so
that actually its radius coincides perfectly with the main attractor P[38]:

In (= 38.00

\ / 'proton /

Archeology is based mostly on the interpretation of written documents,
such as ancient books and manuscripts, inscriptions on ancient build¬
ings. The main problem is that an inscription can also be made a long
time after (even hundreds of years later) a temple was built. And it is
well known that at all times, triumphant conquerors rewrote history.

But, how can we know, when and how an ancient construction was
built? Andrey Sklyarov 1 , a Russian researcher, has shown that we can
discover all this by studying the “language of the stones”.

Probably, even the Great Pyramid of Giza (GPG) doesn’t contain
any “chamber of antique knowledge”. Rather the pyramid by itself
embodies this knowledge. And the giant scale of this construction points
to the importance the builders attributed to this knowledge.

I suspect that no explaining script will be found, because only stones
can survive time. “Mankind fears Time, but Time fears the Pyramid.”
Considering the FF as feature of the number continuum, perhaps this
Arab proverb refers to its timelessness.

Let’s look at the GPG as a message that the builders treated as the
most important knowledge they wanted to conserve for all the future
generations.

Egyptologists 2 believe that the GPG was built as a tomb for the
4 th Dynasty Egyptian pharaoh Khufu (Cheops). The completed design
dimensions, as suggested by Petrie’s survey and later studies, are es¬
timated to have originally been 280 Egyptian Royal cubits (146.5 m)
high by 440 cubits (230.4 m) long at each of the four sides of its base.
In this case, the ratio of the perimeter to height of 1760/280 Egyptian
Royal cubits would equate to 27r with the well-known approximation of
7r as 22/7 (p. 13).

However, these are hypotheses and not facts. Furthermore, these
estimations are based on a geometric model of an ideal pyramid that
does not coincide with the realities of the GPG.

In this book we don’t develop hypotheses about what could be the
original design of the GPG. Let us analyze only facts in the meaning of

1 Sklyarov A. The Myth about Flood: calculations and reality, www.lah.ru

2 Rainer Stadelmann: Die agyptischen Pyramiden. Vom Ziegelbau zum Weltwun-
der. Philipp von Zabern Verlag, 1997.

Nothing is Artificial in the Universe

what we can measure and let us assume that the builders did consider
the consequences of erosion and other destructive forces.

Actually, the length of the base is not 230 m, but 225 m (like the
Pyramide of the Sun in Teotihuacan), and it fits perfectly with the main
attractor of stability E[34]:

225 m

^electron

34.00

Considering late expansions, removals and repairs, the height 1 of the
actual pyramid body is not 146 m, but 137 m like the Chephren pyramid
and coincides well with the main attractor P[41]:

137 m \

-^proton /

41.01

Considering only realities, the GPG is what it is — a square frustum,
like many other pyramids around the world including the great Mexican
pyramids. Consequently, the existence of the missing pyramidion could
turn out to be a dispensable assumption. If there was some construction
on the top or not — we cannot measure it anymore.

The current volume of the GPG is estimated to be 2.5 million cubic
meters that coincides with the main attractor P [123]:

2.5 • 10 6 m 3
A 3

/x proton

123.02

By the way, P [123] = 3P[41]. This means that the volume V of the
GPG equals to the cube of its actual height h:

V = h 3

That’s an amazing fact. Here we don’t analyze the sizes and the geom¬
etry of the chambers, floors and shafts. This will be done in another
book. However, from the point of view of Global Scaling, it seems not
surprising that the GPG appears as an example of proton and electron
stability.

By the way, the Hagia Sophia’s giant dome rests on four arches,
which are in turn supported by a series of columns and semi-domes.
If any of the supports fails, the dome would collapse. To understand

1 Zahi Hawass. The Treasures of the Pyramids. White Star Publ., Torino (Italy),
2003.

Nothing is Artificial in the Universe

the potential danger to the building, earthquake specialists have built a
scale model 1 and performed tests. The team believes that if the model
will be damaged by a simulated earthquake, the actual building might
be damaged in the same way by a real earthquake.

Talking about modelling, it is a good occasion to learn even more
about Global Scaling. One of the principal consequences of the explicit
nonlinearity of the FF is that any downscaled real model can not fully
simulate the behavior of the original. It is so, because the position in
the FF of the model is compellingly different from the position of the
original. If the dimension of the original coincides with a main attractor,
it is not automatically valid for the model. Consequently, a downscaled
model can be damaged by stress while the original will not be affected.

Now you can understand that correctly scaled modelling is not pos¬
sible without knowledge of the FF. Here arises the question if there are
some scaling factors providing for best similarity of the model with the
original.

First we can suggest: If the scaling factor of the model is chosen to
be an integer potency of Euler’s number, some of the model’s features
will coincide with the original. In this way, if one of the original’s
measurements coincides with an attractor, the corresponding model’s
measurement will also coincide with a similar attractor. However, the
similarity with the original will be very relative, because it concerns
only one physical quantity.

For example, if the radius or height of the original matches with
a main attractor and the model is downscaled by a integer potency
of Euler’s number, other model properties like volume or mass do not
automatically match with main attractors.

Naturally, this is valid also for wind tunnel modelling. Only a few
objects can be investigated in a wind tunnel without scaling. For air¬
planes or buildings scaled down models are used. Wind tunnel models
of aircraft and spacecraft are designed to extract aerodynamic data for
analysis of their full-scale counterparts at specified flight conditions. For
example, at the National Transonic Facility at the NASA Langley Re¬
search Center, 2.7-percent scale models of the Boeing 777 airplane are
mounted to sting support systems for testing at transonic speeds. 2

1 Aliberti L. et al. New contributions on the dome of the Pantheon in Rome: com¬
parison between the ideal model and the survey model. The International Archives
of the Phot ogrammet ry. Remote Sensing and Spatial Information Sciences, vol. XL-
5/W4, 2015.

2 Chambers J. R. Modeling Flight. The Role of Dynamically Scaled Free-Flight
Models in Support of NASA’s Aerospace Programs. NASA Publ., 2010.

Nothing is Artificial in the Universe

It isn’t difficult to understand that a scaling factor of 0.027 moves the
downscaled model in a region of the FF that has nothing in common
with the position of the original, because ln(0.027) = —3.612 doesn’t
match with any main- or first layer attractor like ±1/2, ±1/3, ±1/4.

In the consequence, some aerodynamic characteristics of the origi¬
nal can be simulated, but the mechanical behavior including internal
resonance can’t be tested at all. And so, the test flight of the virgin
prototype is still an unavoidable risk, because it will necessarily expose
some unpredictable problems which did not appear in the tests of the
real model. Therefore, it is always worth to study nature and to learn
how the metric characteristics of an established process or structure are
distributed in the FF.

Let me return for a moment to the roots of the Fundamental Fractal.
Primarily, the FF defines the distribution of frequency ratios which do
not support resonance. In this way, any process can avoid destabilizing
internal resonance.

In terms of arithmetics, resonance is a question of divisibility without
rest. For example, if the duration of two cycles is 3 and 4 seconds
respectively, they interact every 12 seconds, because the whole number
12 is divisible by 3 and 4 without rest.

This is valid for all rational numbers being whole number ratios.
However, it is not valid for irrational numbers if they are transcendental,
because there is no algebraic equation describing them.

Therefore, real transcendental ratios in general exclude resonance,
and rational potencies of Euler’s number also exclude resonance regard¬
ing all derivatives of a process.

It is important to realize that divisibility means the division of a
set into parts of equal quantity of elements. Only this multiplicative
definition of division corresponds with the meaning of a frequency. For
example, 8/4 = 2 means 8 = 2±2±2±2. In this case, the frequency
is equal to 4 and the duration of one cycle is 2. The same quantity can
be divided in non equal parts, for instance 8 = 5 ± 3. In this case it is
not possible to define a frequency.

From this point of view, we can understand the origin of the FF in
terms of whole numbers and their divisibility and therefore, in terms of
sets and their cardinality (number of elements).

In general, any natural number can be interpreted as the cardinality
of a set. As we already know, transcendental numbers can be approx¬
imated well by rational numbers, and in some cases also by natural
numbers. This is also valid for the rational powers of Euler’s number.
For example, the natural number 20 is a good approximation of e 3 , 90

Nothing is Artificial in the Universe

is a good approximation of e 4 ' 5 , and the transcendental number e 9 can
be approximated well by the natural number 8103.

Coinciding with attractors of stability, those natural numbers rep¬
resent cardinalities of stable sets in the sense that these cardinalities
prevent any set from destabilizing resonance.

Cardinalities are natural numbers, and the reference unit is the num¬
ber 1. Therefore, for cardinalities, we write the FF-attractors in square
brackets, for example [9] = e 9 , without E or P.

The interpretation of the FF as distribution of cardinalities of stable
sets independent of the nature of the set significantly extends the field of
possible applications, including statistics, for example in economy and
finances.

As the value of money is represented by real numbers, we can apply
the FF also for analyzing the money circulation. In this way we can
recognize amounts of higher or lower degree of stability.

Let’s analyze the established banknotes where each represents a set
(amount) of currency with a fixed cardinality (face value).

Since the face values of the banknotes are adapted to the decimal
numeral system, only a few of them coincide with main FF-attractors
of stability. Table 4 on the next page shows that only the face values
20 and 50 fulfill this criterion (not counting the trivial case of the face
value 1). Which consequences could it have, what do you think?

Provided sufficient liquidity, it could mean that the probability to
have banknotes of €1 20 is higher than the probability to have banknotes
of €1 10, for example. This circumstance should affect the quantity of
banknotes needed in global circulation. This quantity should signifi¬
cantly depend on the face value.

In fact, statistical data of the Bank of England 1 show that there are
nearly three times more £20 bills in circulation than, for example, £\0
bills. Instead, the European Central Bank 2 tells that the €1 50 bill holds
the absolute championship in European domestic circulation.

The US Federal Reserve 3 statistics shows that the $20 bill is the
most distributed in domestic circulation. At the same time, the $100
bill is produced produced in incomparably higher quantities, seven times
more than the number of $20 bills, but this giant amount of $100 bills
doesn’t participate in the domestic circulation. How could it be?

1 Banknote Statistics. Bank of England. 2018, www.bankofengland.co.uk

2 Banknote Circulation. European Central Bank. 2018, www.ecb.europa.eu

3 Currency in Circulation: Volume. Currency and Coin Services. Board of Gov¬
ernors of the Federal Reserve System. 2018, www.federalreserve.gov

Nothing is Artificial in the Universe

FACE VALUE C

LN (C)

LN (C) - FF

[ 0 ]

0.69

[ i ;- 3 ]

0.03

1.61

I 2 ;- 3 ]

- 0.06

2.30

[ 2 ; 3 ]

- 0.03

3.08

[ 3 ]

0.08

3.91

[ 4 ]

- 0.09

100

4.61

[ 5 ; 3 ]

- 0.06

200

5.30

[ 5 ; 3 ]

- 0.03

500

6.21

[ 6 ; 4 ]

- 0.04

Table 4: Face values of banknotes and their correspondence with FF-
attractors. Not counting the trivial case of the face value 1, only the face
values 20 and 50 coincide with main attractors.

According to a report in The Atlantic 1 , $100 bills are a preferred
medium of exchange for facilitating clandestine transactions, and for
storing illicit and untaxed wealth. These include the illegal trade in
drugs, arms and human trafficking as well as amounts of ’unreported’
income. An overwhelming majority of the $100 bills comes from the
Federal Reserve Cash Office in New York City, which handles the bulk
of foreign shipments of US currency. A typical shipment is a pallet
containing 640,000 such bills, or $64 million. By the way, this amount
is close to the main attractor [18]: ln(6.4 • 10 7 ) = 17.97.

Indeed, the $100 bill does not coincide with a main attractor of
stability and therefore, the quantity of $100 bills should not exceed
the quantity of $20 or $50 bills which coincide with main attractors.
Consequently, the $100 bill imbalance shows that there is a black market
that can destabilize the circulation. In this way, the knowledge about
the FF can be applied in economics, and opens the possibility for having
information about the financial “state of health”.

Now let’s come back to mathematics. Talking about divisibility of
numbers as resonance condition, we can apply this principle to the loga¬
rithms as well.

x Matt Phillips. $100 bills make up 80% of all U.S. currency — but why? The
Atlantic , 21 November 2012, www.theatlantic.com

Nothing is Artificial in the Universe

In general, logarithms are real numbers, and the logarithms of main
attractors are whole (integer) numbers. These integer logarithms can
be of higher or lower divisibility without rest and therefore, the corre¬
sponding attractors have more or less interscalar resonance.

Consequently, a main attractor of higher divisibility, for example
P [54], has more interscalar connections with other main attractors than
a main attractor of lower divisibility, for example P [51].

Prime number attractors form the basis of interscalar connections
because of their non-divisibility. The integer potencies of primes define
the position of a main attractor in the interscalar hierarchy. In this
sense, square, cubic or higher potencies of prime numbers occupy key
positions.

For example, the Sun (p. 31) occupies the main attractor E[49] that
is the square of the prime 7. The human zygote (p. 41) occupies the main
attractor P[27] that is the cube of 3. By the factor 3 it is connected with
the main attractor P[81], the scale of the Galactic Core. The “master
planets” Jupiter and Saturn, the “master frequencies” of Theta brain
activity (p. 39) and the pituitary “master gland” (hypophysis, p. 41)
occupy the main attractor P[54] that is the double cube of 3. The orbit
of Venus (p. 30) occupies the main attractor E[54].

Divisible by 12 numbers define local islands of maximum divisibil¬
ity 1 . For example, the number 60 is divisible by 12, but also by 2,
3, 4, 5, 6, 10, 15, 20, 30 whereas the neighboring numbers 59 and 61
are prime. In consequence, the divisible by 12 attractors of proton and
electron stability define channels of maximum interscalar connectivity.

Let us look at some divisible by 12 main attractors and the scales
they correspond to. For example, the average size of eukaryotic cells
corresponds with the main attractor P[24]. The next divisible by 12
attractor P[36] defines the range of the human body size (pp. 46-47).
The average human adult relaxing heart rate (p. 26) corresponds with
the main attractor E[48]. The mass of the pineal gland (p. 42) coincides
with the attractor E[60], and the weight of the spleen corresponds with
the attractor P[60].

Jupiter’s surface gravity acceleration (p. 31) meets the main attrac¬
tor P[72] that defines also the scale of the Oort cloud (0.5 light years),
the hypothetic boundary of the solar system. The radius of our Galaxy
coincides with the main attractor P[84] and finally, the scale of the
observable universe (light horizon) coincides with the attractor P[96].

1 Shnoll S. E. Cosmophysical factors in stochastic processes. American Research
Press, 2012, pp. 400-404

Nothing is Artificial in the Universe

As inherent feature of the number continuum, the FF rules the course
of any process and forms all the structures in the universe, regardless
of their nature and complexity.

Global Scaling suggests that there is nothing artificial in the uni¬
verse. All the technology developed by humanity and other civilizations
follows the same FF like anything in the universe. Applying Global
Scaling, you will see the world with new eyes and explore space and
time in a way that hasn’t been done before.

Global Scaling leads to an interscalar view of the world that could
be a new scientific paradigm. Showing the complex connection of pro¬
cesses at very different scales in the universe, Global Scaling explains
mathematically how subatomic and galactic scales are directly related
to life as a cosmic phenomenon.

The nature of life isn’t competitive struggle. Life is interscalar com¬
munication and cooperation.

In the interscalar view, we are not isolated beings, but we are embed¬
ded in the solar system, we are an integral part of cosmic life. This is not
a poetic phrase, but a scientifically demonstrable fact. This knowledge
should be considered in medicine, but also in scholastic education.

Science does not progress because of brilliant minds constantly cre¬
ating better and better theories. The theory is not the motor of science,
rather it is the discovery itself. Nobody can foresee a discovery, it comes
unexpectedly and forces the scientist to develop the theory.

Surely, there is a huge field of research where further astonishing
discoveries are awaiting us.

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2, 99-105, 2018.

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Acknowledgements

Many have come into this century
to develop arts and sciences,
sow the seeds of a new culture
that will blossom, unexpected, just
when the power is believing it has won.

Giordano Bruno

There are many individuals to whom I am deeply grateful. First of
all, I am grateful to my parents who always created best conditions for
my studies. With gratitude I remember my high school teachers who
enabled me to incorporate some of their lessons.

I am grateful to my country that gave me the possibility to study
for free at one of the most famous universities.

I am thankful to Vera Reutova for giving me two beautiful children.
Veronika, Erwin and my brother Uwe are always in my heart. They
are supporting me and were fighting for me when the power tried to
humiliate and destroy me.

I am infinitely grateful to Leili Khosravi. Her love kept me alive
when I was locked up for 21 months with 4 prisoners in a completely
dark 2 meter small all-metal cell. Leili shares her life with me giving all
the light I need to continue my work.

I am grateful to my teachers, colleagues and friends at the Saint
Petersburg State University, at the Moscow State University, at the
Russian Academy of Sciences and the Volgograd State University of
Technology. I am especially grateful to Oleg Kalinin, Simon Shnoll,
Victor Panchelyuga and Valery Kolombet, Maria Kondrasheva and Irina
Zaychkina, Yury Vladimirov, Dmitry Pawlov, Alexander Beliaev, Ale¬
xey Petrukhin and Alexander Polovinkin. They guided my scientific
research and made possible the discovery I share with you in this book.

I’m grateful to my graduates and friends: Michael Kauderer and
Ulrike Granogger, Ronny and Katja Kircheis and many others who con¬
tinued to work in the held of Global Scaling, even when it was already
defamed by the mass media.

Especially I am grateful to Urs Biihler and Marcel Bauer who found¬
ed the HealthBalance Centre in Uzwil, Swizerland and made possible
the application of my research in healing and architecture.

Acknowledgements

I am grateful to Hans-Joachim and Kathe Ehlers, who gave me the
possibility to continue my research and to publish it.

I am grateful to my friends in the Community of Living Ethics,
especially to Giuseppe Campanella, Marina Bernardi, Gabriella Fini,
Paola Bucetti and many others who continuously support my work.

Many thanks to Dmitry Rabounsky, Adreas Ries and Felix Scholk-
mann who made possible the publication of this book.

About the Author

All truth goes through three stages:

First it seems ridiculous,

then it is fought,

after all, it is self-evident.

Arthur Schopenhauer

Hartnrut Muller, born 1954 in Hildburghausen, GDR, studied philo¬
sophy and natural sciences at the Saint Petersburg (Leningrad) State
University. During a research assistantship, he was also trained in epi¬
stemology of scientific research, applied mathematics, particle physics
and engineering science.

From 1978 to 1990 Hartnrut Muller was teaching philosophy and
epistemology of research in engineering sciences at the Volgograd State
University of Technology, where he developed Global Scaling methods of
analysis and optimization of technology, archived at the Soviet Institute
for Scientific and Technical Information.

After his return to Germany, Hartmut Muller became editor of the
journal raurn und zeit of the Ehlers publishing house and cofounded
the Global Scaling Research Institute and the non-profit Global Scaling
Association for support of research and education.

Hartmut Miiller is noted for his public commitment to the non¬
military application of scientific research. In 2004, for his research and
commitment to ethics in science he received the Vernadski medal, the
highest recognition of the non-governmental Russian Academic Society.
As a result of a slander campaign, he was charged in 2012 with scientific
fraud and convicted.

Hartmut Muller has published many scientific papers as well as pop¬
ular science articles on Global Scaling in particle physics, astrophysics
and cosmology, geophysics and biophysics, engineering science and ar¬
chitecture. He also published patents on Global Scaling applications.
Today he continues his scientific studies as an independent researcher.

Global Scaling

the fundamentals of

interscalar cosmology

by Hartmyt MuNer

This book is designed as a quick and easy introduction to Interscalar Cosmology that is based on
the discovery of a universal law of probably everything - Global Scaling.

The discovery of Global Scaling is the result of an interdisciplinary research that has inspired the
author for more than 35 years.

Global Scaling leads to an interscalar view of the world that could become the new scientific
paradigm. Showing the complex connection of processes at very different scales in the universe.
Global Scaling explains mathematically how subatomic and galactic scales are directly related to
life as a cosmic phenomenon.

Global Scaling suggests that there is nothing artificial in the universe. All the technology
developed by humanity and other civilizations follows the same fundamental fractal of space-time
just like everything else in the universe. It may well be that Global Scaling is the rediscovery of an
ancient advanced knowledge.

The reader of this book will find new answers to many questions concerning the nature of the
universe and the meaning of life.

A huge field of research with new astonishing discoveries awaits us.

9 780998 189406 >

New Heritage Publishers

Brooklyn, New York, 2018

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